# An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution

Authors: Michael Milgram
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann’sξs function and thus ζs indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates ζs anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, both an analytic expression for ζσ+it, everywhere inside the asymptotic t⟶∞ critical strip, as well as an approximate solution can be obtained, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of ζσ+it for different values of σ and equal values of t; this is illustrated in a number of figures.


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## Body

## 1. Introduction

The Riemann zeta functionζs is well known to satisfy a functional equation, and many representations, both integral and series, have been developed over the years for both ζs and its avatar ξs. Additionally, and more significantly, at least two independent contour integral representations are known ([1], Section (1.4)) and ([2], equation (7)) either of which could be utilized as the primary definition, from which many of the properties of ζs can be derived. On the contrary, it has been long-ago proven that ζs does not satisfy a differential equation of quite general form ([3] (and citations therein)); also known are three integral equations which ζsdoes satisfy: [4], equation (1.6), [5], equation (3.29), and [6], equation (1.5) with α=0. In this work, a series representation of ξs in terms of generalized integro-exponential functions (Esz) is coupled with a contour integral representation of these functions, to obtain a new integral equation satisfied by ξs and, equivalently, ζs. This equation has several remarkable properties:(i)
Through the action of an integral operator, the value ofζs1 (the “dependent”) anywhere in the complex plane can be determined with respect to the properties of ζs2 (the “master”) on a line ℜs2=constant anywhere else in the punctured s1≠s2 complex plane(ii)
The transfer function that mediates the aforementioned action is a simple rational polynomial function ofs1 and is therefore quite amenable to analysis(iii)
The onlys1 dependence resides in the transfer functionIn this work, after initially listing some definitions and lemmas (Section c), a derivation of the integral equation over a bounded region of thes2-plane (Section 3) is given, and its analytic continuation over the remainder of the punctured s1≠s2 complex plane (Section 4) is obtained. In Section 5, various representations are then used to obtain a few simple integrals involving ζs2, corresponding to special values of s1; it is also demonstrated that the simple integrals under consideration are convergent.Following these preliminaries, the general form of the transfer function is carefully presented in a series of Appendices. Based on a particularly useful property of the transfer function presented in Appendix C—it closely mimics a Dirac delta function on thes2 line— a reasonably accurate model for ζs1 is established, which yields reasonably accurate estimates of the asymptotic nature of ζs1 inside the critical strip (Section 6). In consequence, it is shown that, asymptotically(1)ζσ+it∼t1/2−σ/2×loglogtlogt+⋯,and, within the confines of the model, it is both proven that ζs1≠0 if ℜs1≠1/2, and explained, in transparent terms (Section 7), why this happens (does not happen?). Graphical examples and comparisons are given, demonstrating the accuracy of a predicted universality between the real and imaginary components of ξσ+it at equal values of t but different σ. In Section 8, two well-accepted properties of ζs, notably Universality ([7], Chapter XI) and Lindelöf’s hypothesis ([1], Section 9.2) are discussed in the context of the results reported here, and suggestions are made that merit future consideration. Section 9 discusses requirements needed to improve the rigour of the model developed here.


## 2. Preamble

### 2.1. Notation

Throughout,I use s=σ+it to denote the independent variable defining ζs. In those instances, where dependence on, for example, t is the focus, I will sometimes just shorten, for example, Mc,s,v⇒Mt,v, for typographical brevity and clarity. Throughout, subscripts ‘R’ and ‘I’ refer to the real and imaginary components of whatever they are attached to. Much use is made of Riemann’s ξ function, defined by(2)ξs≡s−1π−s/2ζsΓ1+s2,as well as(3)ϒs≡ζsΓs2π−s/2,both of which satisfy(4)ξs=ξ1−s,(5)ϒs=ϒ1−s.Throughout,k,N=0,1,2,….


### 2.2. Definitions and Lemmas

Definition.
Polar form means that a complex functionfs is written as expiθsfs, where θs≡argfs.(i)
The following asymptotic limits ([8], equation (5.6.9); see also [7], equation (4.12.2)) will be required:(6)limt⟶∞Γσ2+it2≈2πσ24+t24σ−1/4exp−πt4,and the special case(7)limt⟶∞Γ12±it2≈2πexp−πt4.Remark:.
Approximation (6) was tested numerically and found to be remarkably accurate for even modest values of t when 0<σ<1≪t.
(ii)
From the functional equation ofζs with (6), for large values of t,
(8)
ζ
i
t
∼
ζ
1
−
i
t
t
2
π
.Lemma 1.
(9)
arg
ζ
1
2
+
i
t
+
arg
Γ
1
4
+
i
t
2
−
t
log
π
2
=
±
k
π
.
Well-known ([7], equation (4.17.2)) with k=0, this is sometimes referred to as the Riemann–Siegel identity. A simple derivation with k≠0 follows.Proof.
Expandξσ+it about σ=1/2 at constant t giving(10)limσ⟶1/2ξσ+it=ξ12+it+∂∂σξσ+itσ=1/2σ−12+⋯,and find the imaginary part from (2), after rewriting ζs and Γs/2 in polar form:(11)ℑξ12+it≡ℑlimσ⟶1/2ξσ+it=−4t2+1sinargζ1/2+it+argΓ1/4+it/2−t/2lnπΓ1/4+it/2ζ1/2+it8π1/4.
Becauseξ1/2+it is known to be real, (11) vanishes, and (9) follows immediately. QED.Remark.
Apply the identity(12)argζ12+it=ℑlogΓ1/2+it+N.to (9) to obtain Backlund’s formula which counts the discontinuities (not zeros) of argζ1/2+it (see [9], Section 9). This is consistent with Theorem 1 in [10].(iii)
Define(13)ϕs≡argζs+argΓs2−12tlogπ=argϒs,(14)θv=argΓ32+iv+argζ1+2iv−vlnπ,(15)Φt≡argζit+argΓit2−t2logπ,(16)αt≡argζ12+it,(17)βt≡argζ′12+it.
However, for brevity, expressions can be abbreviated as follows:(18)cosΦt≔cosΦ.


## 2.1. Notation

Throughout,I use s=σ+it to denote the independent variable defining ζs. In those instances, where dependence on, for example, t is the focus, I will sometimes just shorten, for example, Mc,s,v⇒Mt,v, for typographical brevity and clarity. Throughout, subscripts ‘R’ and ‘I’ refer to the real and imaginary components of whatever they are attached to. Much use is made of Riemann’s ξ function, defined by(2)ξs≡s−1π−s/2ζsΓ1+s2,as well as(3)ϒs≡ζsΓs2π−s/2,both of which satisfy(4)ξs=ξ1−s,(5)ϒs=ϒ1−s.Throughout,k,N=0,1,2,….


## 2.2. Definitions and Lemmas

Definition.
Polar form means that a complex functionfs is written as expiθsfs, where θs≡argfs.(i)
The following asymptotic limits ([8], equation (5.6.9); see also [7], equation (4.12.2)) will be required:(6)limt⟶∞Γσ2+it2≈2πσ24+t24σ−1/4exp−πt4,and the special case(7)limt⟶∞Γ12±it2≈2πexp−πt4.Remark:.
Approximation (6) was tested numerically and found to be remarkably accurate for even modest values of t when 0<σ<1≪t.
(ii)
From the functional equation ofζs with (6), for large values of t,
(8)
ζ
i
t
∼
ζ
1
−
i
t
t
2
π
.Lemma 1.
(9)
arg
ζ
1
2
+
i
t
+
arg
Γ
1
4
+
i
t
2
−
t
log
π
2
=
±
k
π
.
Well-known ([7], equation (4.17.2)) with k=0, this is sometimes referred to as the Riemann–Siegel identity. A simple derivation with k≠0 follows.Proof.
Expandξσ+it about σ=1/2 at constant t giving(10)limσ⟶1/2ξσ+it=ξ12+it+∂∂σξσ+itσ=1/2σ−12+⋯,and find the imaginary part from (2), after rewriting ζs and Γs/2 in polar form:(11)ℑξ12+it≡ℑlimσ⟶1/2ξσ+it=−4t2+1sinargζ1/2+it+argΓ1/4+it/2−t/2lnπΓ1/4+it/2ζ1/2+it8π1/4.
Becauseξ1/2+it is known to be real, (11) vanishes, and (9) follows immediately. QED.Remark.
Apply the identity(12)argζ12+it=ℑlogΓ1/2+it+N.to (9) to obtain Backlund’s formula which counts the discontinuities (not zeros) of argζ1/2+it (see [9], Section 9). This is consistent with Theorem 1 in [10].(iii)
Define(13)ϕs≡argζs+argΓs2−12tlogπ=argϒs,(14)θv=argΓ32+iv+argζ1+2iv−vlnπ,(15)Φt≡argζit+argΓit2−t2logπ,(16)αt≡argζ12+it,(17)βt≡argζ′12+it.
However, for brevity, expressions can be abbreviated as follows:(18)cosΦt≔cosΦ.


## 3. LeClair’s Representation

In a recent work, LeClair ([11], equation (15)) has obtained the following series representation of Riemann’s ξ function:(19)ξs=s−1π∑n=1∞n2E−s/2πn2−πs∑n=1∞n2Es−1/2πn2+4π∑n=1∞n2e−πn2,where generically, Esz is the (generalized) “Exponential Integral,” a limiting case of what is elsewhere [12] referred to as the “Generalized Integro-Exponential Function.” Furthermore, it has been shown ([13], equation (3)) that the infinite sum in (19) can be readily evaluated as follows:(20)4π∑n=1∞n2e−πn2=π1/42Γ3/4.In his work, LeClair truncated the sum(s) atN terms, referred to the result as an “approximation” and proceeded to obtain approximations to the location of the zeros on the critical line on that basis. Here, the sums are treated as an infinite series representation of ξs, and hence an identity because each series is easily shown to be convergent due to the asymptotic property of Esz (see [12], equation (2.25)). Similar, but inequivalent series representations will be found in Paris ([14], equation (1)), Patkowski ([4], equation (1.20)), and elsewhere. From [12], some useful integral and contour integral representations of the function Esz are(21)Esz=zs−1Γ1−s,z,(22)=∫1∞v−sexp−zvdv,(23)Esz=12πi∫c−i∞c+i∞Γ−vzvs−1−vdv.In (21), Γ1−s,z is the (upper) incomplete gamma function, and (22) provides the fundamental definition of Esz. The result (23) is given in [12] (equation (2.6a)). Here, the integration contour, originally defined to enclose the real axis v≥0 as well as the singularity at v=s−1 in a clockwise direction, has been converted into the line c<ℜs−1 because the integrand vanishes as v⟶±i∞. This paper investigates the application of (23) to (19).


### 3.1. A Novel Integral Representation

The following, focuses on the limited range0≤σ≤1, where s=σ+it, in which case, applying (23) to the first term on the right-hand side of (19) yields(24)πs−1∑n=1∞n2E−s/2πn2=s−12i∫c1−i∞c1+i∞Γ−vπv∑n=1∞n2+2v−s/2−1−vdv,(25)=s−12i∫c1−i∞c1+i∞Γ−vπvζ−2v−2−s/2−1−vdv,where the interchange of integration and summation in (24) is justified if c1<−3/2 since the sum converges under this condition. The general requirement that c1<−σ/2−1 imposes the effective constraint c1<−3/2 for 0≤σ≤1: the contour may be shifted leftwards with impunity since there are no singularities in that direction. Similarly, under the transformation s⟶1−s, we find the following expression for the second right-hand side term of (19):(26)−πs∑n=1∞n2Es−1/2πn2=−s2i∫c2−i∞c2+i∞Γ−vπvζ−2v−2−3/2+s/2−vdv,again valid for c2<−3/2 if σ>0. In the case that c1=c2=c, where c<−3/2, and focussing on 0≤σ≤1, we find the integral equation:(27)ξs=iss−1/2π∫c−i∞c+i∞ξv+3v+33−s+vs+2+vdv+i2π∫c−i∞c+i∞ξv+3v+3s+2+vdv+π1/42Γ3/4,which, in terms of ζs, can be rewritten as(28)s−1ζsΓs2+1π−s/2=2iss−12∫c−i∞c+i∞Γ−vπvζ−2v−2−3+s−2vs+2+2vdv−i2∫c−i∞c+i∞Γ−vπvζ−2v−2s/2+1+vdv+π1/42Γ3/4.Notice that with a simple application of the recursion formula forΓ−v, the numerator in (28) can be written in terms of ϒ−2v−2 (see (3)). Alternatively, under a simple change of variables, with the same condition c<−3/2 and the same range of σ, (28) becomes(29)ξs=π1/42Γ3/4−Js,c,where(30)Js,c=∫−∞∞πc−ivζ−2c+2iv−2Γ−c+ivMc,s,vdv,(31)Mc,s,v=−2s−12/2+2iv−2c+5/4−2s−12/4+2i2iv−4c+5/4v+4c+5/42.This result is worthy of a few comments:(i)
Its form is almost (exception: see [4], equation (1.18)) unique among representations of ζs. Usually s-dependence that is formally embedded inside an integral or series representation of ζs appears either as an exponent or buried inside the argument of a transcendental function; here, s-dependence exists only in the form of a coefficient in a simple rational (polynomial) function. This augers well for further analysis and leads to some surprising predictions;(ii)
(27) and (28) present a prescription, in the form of an integral transform, for the value of ξs or ζs anywhere in the complex strip 0≤ℜs≤1, that depends only on its values on a vertical line in the complex plane corresponding to c<−3/2. The region in which (29) is valid is labelled “I” in Figure 1(b) and delineated as everything to the left of the point at v=−1.5 in Figure 1(a);(iii)
In (29) and (30), the function Mc,s,v acts as a transfer function between ζv on the line ℜv=−c−1 and ζs elsewhere in the complex plane, through the medium of the integral operator (30). Its properties will be of interest in studying (29) for varying values of the parameters c and s;(iv)
Based on (4), it is possible to show ([15], Theorem 1) that (28) can be cast into a simple closed (Cauchy) contour integral for ξs.In the following sections, simple choices ofs and c,c1, and c2 will be applied to (29).Figure 1
The various regions of validity of the analytic continuations developed in Section4 both (a) in the complex v-plane and (b) in the cross section of a complex plane cut by the real σ,c plane at t=0. (a) The location of the contour of integration (black arrow at ℜv<−3/2) extending from −i∞toi∞ as well as the various poles. The fixed pole at v = −3/2 corresponds to the singularity belonging to ς−2v−2. The green and red poles correspond to singularities of the integrand that depend on the value of s acting as a parameter. The coloured arrows show the motion of corresponding poles as ℜs increases from zero to one if t > 0. The colours correspond to colours used in Figure 1(b). (b) The various regions where different continuations of (29) apply, projected onto the t = 0 plane. Five distinct regions are bounded by (coded and coloured) lines c = −3/2 (blue, solid), c=−σ/2−1 (red, dash-dot), and c=σ/2−3/2 (green, dash-dot) when 0≤σ≤1. The bounding regions extend vertically out of the plane of the figure when t≠0 because the contour used in (23) is chosen to be a straight line, vertical in the complex v-plane (c is a constant). The dotted line corresponds to the interesting case c = −5/4.
(a)
(b)


## 3.1. A Novel Integral Representation

The following, focuses on the limited range0≤σ≤1, where s=σ+it, in which case, applying (23) to the first term on the right-hand side of (19) yields(24)πs−1∑n=1∞n2E−s/2πn2=s−12i∫c1−i∞c1+i∞Γ−vπv∑n=1∞n2+2v−s/2−1−vdv,(25)=s−12i∫c1−i∞c1+i∞Γ−vπvζ−2v−2−s/2−1−vdv,where the interchange of integration and summation in (24) is justified if c1<−3/2 since the sum converges under this condition. The general requirement that c1<−σ/2−1 imposes the effective constraint c1<−3/2 for 0≤σ≤1: the contour may be shifted leftwards with impunity since there are no singularities in that direction. Similarly, under the transformation s⟶1−s, we find the following expression for the second right-hand side term of (19):(26)−πs∑n=1∞n2Es−1/2πn2=−s2i∫c2−i∞c2+i∞Γ−vπvζ−2v−2−3/2+s/2−vdv,again valid for c2<−3/2 if σ>0. In the case that c1=c2=c, where c<−3/2, and focussing on 0≤σ≤1, we find the integral equation:(27)ξs=iss−1/2π∫c−i∞c+i∞ξv+3v+33−s+vs+2+vdv+i2π∫c−i∞c+i∞ξv+3v+3s+2+vdv+π1/42Γ3/4,which, in terms of ζs, can be rewritten as(28)s−1ζsΓs2+1π−s/2=2iss−12∫c−i∞c+i∞Γ−vπvζ−2v−2−3+s−2vs+2+2vdv−i2∫c−i∞c+i∞Γ−vπvζ−2v−2s/2+1+vdv+π1/42Γ3/4.Notice that with a simple application of the recursion formula forΓ−v, the numerator in (28) can be written in terms of ϒ−2v−2 (see (3)). Alternatively, under a simple change of variables, with the same condition c<−3/2 and the same range of σ, (28) becomes(29)ξs=π1/42Γ3/4−Js,c,where(30)Js,c=∫−∞∞πc−ivζ−2c+2iv−2Γ−c+ivMc,s,vdv,(31)Mc,s,v=−2s−12/2+2iv−2c+5/4−2s−12/4+2i2iv−4c+5/4v+4c+5/42.This result is worthy of a few comments:(i)
Its form is almost (exception: see [4], equation (1.18)) unique among representations of ζs. Usually s-dependence that is formally embedded inside an integral or series representation of ζs appears either as an exponent or buried inside the argument of a transcendental function; here, s-dependence exists only in the form of a coefficient in a simple rational (polynomial) function. This augers well for further analysis and leads to some surprising predictions;(ii)
(27) and (28) present a prescription, in the form of an integral transform, for the value of ξs or ζs anywhere in the complex strip 0≤ℜs≤1, that depends only on its values on a vertical line in the complex plane corresponding to c<−3/2. The region in which (29) is valid is labelled “I” in Figure 1(b) and delineated as everything to the left of the point at v=−1.5 in Figure 1(a);(iii)
In (29) and (30), the function Mc,s,v acts as a transfer function between ζv on the line ℜv=−c−1 and ζs elsewhere in the complex plane, through the medium of the integral operator (30). Its properties will be of interest in studying (29) for varying values of the parameters c and s;(iv)
Based on (4), it is possible to show ([15], Theorem 1) that (28) can be cast into a simple closed (Cauchy) contour integral for ξs.In the following sections, simple choices ofs and c,c1, and c2 will be applied to (29).Figure 1
The various regions of validity of the analytic continuations developed in Section4 both (a) in the complex v-plane and (b) in the cross section of a complex plane cut by the real σ,c plane at t=0. (a) The location of the contour of integration (black arrow at ℜv<−3/2) extending from −i∞toi∞ as well as the various poles. The fixed pole at v = −3/2 corresponds to the singularity belonging to ς−2v−2. The green and red poles correspond to singularities of the integrand that depend on the value of s acting as a parameter. The coloured arrows show the motion of corresponding poles as ℜs increases from zero to one if t > 0. The colours correspond to colours used in Figure 1(b). (b) The various regions where different continuations of (29) apply, projected onto the t = 0 plane. Five distinct regions are bounded by (coded and coloured) lines c = −3/2 (blue, solid), c=−σ/2−1 (red, dash-dot), and c=σ/2−3/2 (green, dash-dot) when 0≤σ≤1. The bounding regions extend vertically out of the plane of the figure when t≠0 because the contour used in (23) is chosen to be a straight line, vertical in the complex v-plane (c is a constant). The dotted line corresponds to the interesting case c = −5/4.
(a)
(b)


## 4. Analytic Continuations

In the following, letc1=c2=c. By shifting the contour (28) into the region c>−3/2 (see Figure 1(a)), various representations are obtained for each of the regions labelled in Figure 1(b), by subtracting 2πi× the residues of the singularities that are transited in the complex v-plane. To enter the region labelled “II” in Figure 1(b), the contour in Figure 1(a) must pass to the right of the fixed pole at v=−3/2, whose residue is given by(32)residueblue=12πi,giving, for region II,(33)ξs=π1/42Γ3/4−1−Js,c.To enter the region labelled III from region II (crossing the red line in Figure1(b)), the corresponding residue (red pole in Figure 1(a))(34)residuered=iξs2π,must be subtracted. The result, valid for region III is(35)0=π1/42Γ3/4−1−Js,c.To cross the green line, either from region II, entering region V, or from region III, entering region IV, the residue(36)residuegreen=isπs−3/2ζ1−sΓ−s/2+1/2s−14,corresponding to the green pole in Figure 1(a) must be subtracted. A similar argument applies for a transition from region V to region IV, the residue (34) must be subtracted. This gives, for region IV,(37)−ξs=π1/42Γ3/4−1−Js,c.In particular, this analysis relates the results applicable to region IV to that for region I, including regions where both representations are unconstrained with respect to0≤σ≤1 and equates the result for region V to that for region III. It is emphasized that the regional definitions in Figure 1(b) exclude the lines, thereby defining the punctured regions s1≠s2 alluded in the Introduction. Of particular interest, is the value at the point s=1/2,c=−5/4, as well as the values along the horizontal (dotted) line s=σ+itc=−5/4 and a vertical line cs=1/2+it. To reiterate, Figure 1(b) is intended to illustrate the projection of a (complex) space s=σ+it defined by a third axis extending out of the plane of the Figure, corresponding to t≠0, onto the (real) plane σ,c where t=0.


## 5. Crossing the Line atc = −3/2

Remark.
Several of the identities in this section are (redundantly) declared to be real and markedℜ. This is done as a reminder that unless so-declared, an attempted numerical computer evaluation of that expression will likely fail because the imaginary part of the identity contains a singularity (e.g., (42)).


### 5.1. The General Casec = −3/2

Disregarding the contour integral representations from which they are obtained, (29), (33), and (35) simply involve several complex functions of two real variables (if t=0) and ought to be amenable to analysis from that viewpoint. Compare (29) and (33), respectively, below and above the horizontal line demarking regions I and II of Figure 1(b) by setting c=−3/2−η in (29) and c=−3/2+η in (33), where 0<η≈0. Note that η is a real variable (there is no cut); since none of the terms in either equation (other than Js,c) depend on c, we are free to choose a convenient value for c, subject to the (regional) conditions under which each equation is valid. We have in region I:(38)ξs=π1/42Γ3/4−Js,−32−η,and in region II(39)ξs=π1/42Γ3/4−1−Js,−32+η.Simple comparison between (38) and (39) suggests that(40)Js,−32−η−Js,−32+η=1.This can be independently verified by a change of variablesv⟶ηv in both integrals, followed by a series expansion about η=0, leading to(41)Js,−32−η−Js,−32+η⟶1π2∫−∞∞11+v2dv=1.As is often done where discontinuities arise in the theory of functions of a real variable, whenη=0 it is conventional to assign half the discontinuity at that point c=−3/2. This is equivalent to deforming the contour in (28) to avoid the fixed pole belonging to ζ−2v−2 at v=−3/2 by including only half the residue at that point.Remark.
This cannot be done for the other poles since they are located bys (complex), rather than v=−3/2 (fixed).


#### 5.1.1. The Caseσ=1/2

The result from (38) and (39) after setting η=0 (see Section 5.1) when s=1/2 is(42)12π∫−∞∞ℑξ1+2iv2iv+1/2vdv=∫−∞∞ℜπ−3/2−ivζ1+2ivΓ3/2+iv2iv+1/2dv=ζ1/2π3/428Γ3/4+π1/42Γ3/4−1/2.This result (compared with ([7], page 204, Lemma β)) can be verified numerically. The need to specify that only the real part of the integrand in (42) is to be used is twofold:(i)
The right-hand side is real and so must be the left-hand side(s).(ii)
Ostensibly, the integrand of (42) appears to be singular at v=0 unless one notes that(43)ζ1±2iv≈∓i2v+γ+⋯,where γ is Euler’s constant. Therefore, the real part of the integrand converges at v=0 and the imaginary part, singular at the origin, integrates to zero by antisymmetry.In general, whens=1/2+it, the left-hand side of both (38) and (39) are real, and so the equivalent of (42) applies. Written in terms of t, (42) generalizes to(44)ξ12+it=−12π3/2∫−∞∞ℜπ−ivζ1+2iv4t2+4iv+1Γ3/2+ivt2+2iv−4v2+1/4dv−12+π1/42Γ3/4.


### 5.2. The Casec = −5/4

In regions III and V, writing (35) in full gives(45)πc∫−∞∞π−ivζ−2c+2iv−2Γ−c+iv2iv−2s2−2c+2s−3−2c+2iv+s−3−2c+2iv−s−2dv=π1/42Γ3/4−1,or, equivalently(46)∫−∞∞2iv−2s2−2c+2s−3ξ−2c+2iv−2−2c+2iv+s−3−2c+2iv−s−2−2c+2iv−3dv=π1/42Γ3/4−1.Settingc=−5/4 in (45) leads to(47)∫−∞∞ζ1/2+2ivπ−5/4−ivΓ5/4+iviv−2s−12/42s−12/8+2v2dv=1−π1/42Γ3/4,a result that is valid for all s (see the dotted line in Figure 1(b)). Since the left-hand side is a function of s and the right-hand side is not, it must be true that the derivative of the left-hand side with respect to s vanishes. Performing this calculation (the integral is convergent, so the derivative and integral operator can be interchanged) yields(48)s−1/2∫−∞∞π−5/4−iv−2−iv+2s−12/4ζ1/2+2ivΓ5/4+iv2s−12/8+2v22dv=0.Compare (47) with Patkowski ([4], Theorem 1).


#### 5.2.1.σ=1/2

Consider (47) in the case s=1/2, which gives(49)∫−∞∞iπ−5/4−iv2vζ12+2ivΓ54+ivdv=1−π1/42Γ3/4,and the integrand apparently has a singularity at v=0. However, a simple expansion of the integrand about that point shows that(50)ℜiπ−5/4−iv2vζ12+2ivΓ54+iv=−ζ1/22π1/4Γ3/4+Ov2,demonstrating that the real part of the integrand in (49) is nonsingular at v=0. However, near v=0, it is similarly shown that the imaginary part of the integrand diverges like v−1 and the imaginary part of the integral vanishes by antisymmetry about v=0.In the cases=1/2+it, (47) can be written in terms of ξ1/2+2iv as(51)1π∫−∞∞ξ1/2+2iviv+t2−t2/2+2v2−1/2+2ivdv=1−π1/42Γ3/4,whose integrand appears to become singular at v=±t/2. Writing the integrand in terms of its real and imaginary parts, and noting that ξ1/2+2iv is real, we find, for the real part(52)4π∫−∞∞ξ1/2+2iv16v2+1dv=1−π1/42Γ3/4,and, for the imaginary part,(53)∫−∞∞vξ1/2+2iv4v2−t216v2+1dv=0.Notice that in the former case, there is not dependence, so in (51), the variable t is in reality a free parameter, consistent with the argument applied to obtain (48). In the latter case, although a singularity exists, the integrand is antisymmetric about v=0, and so the singularities at ±v cancel.


## 5.1. The General Casec = −3/2

Disregarding the contour integral representations from which they are obtained, (29), (33), and (35) simply involve several complex functions of two real variables (if t=0) and ought to be amenable to analysis from that viewpoint. Compare (29) and (33), respectively, below and above the horizontal line demarking regions I and II of Figure 1(b) by setting c=−3/2−η in (29) and c=−3/2+η in (33), where 0<η≈0. Note that η is a real variable (there is no cut); since none of the terms in either equation (other than Js,c) depend on c, we are free to choose a convenient value for c, subject to the (regional) conditions under which each equation is valid. We have in region I:(38)ξs=π1/42Γ3/4−Js,−32−η,and in region II(39)ξs=π1/42Γ3/4−1−Js,−32+η.Simple comparison between (38) and (39) suggests that(40)Js,−32−η−Js,−32+η=1.This can be independently verified by a change of variablesv⟶ηv in both integrals, followed by a series expansion about η=0, leading to(41)Js,−32−η−Js,−32+η⟶1π2∫−∞∞11+v2dv=1.As is often done where discontinuities arise in the theory of functions of a real variable, whenη=0 it is conventional to assign half the discontinuity at that point c=−3/2. This is equivalent to deforming the contour in (28) to avoid the fixed pole belonging to ζ−2v−2 at v=−3/2 by including only half the residue at that point.Remark.
This cannot be done for the other poles since they are located bys (complex), rather than v=−3/2 (fixed).


### 5.1.1. The Caseσ=1/2

The result from (38) and (39) after setting η=0 (see Section 5.1) when s=1/2 is(42)12π∫−∞∞ℑξ1+2iv2iv+1/2vdv=∫−∞∞ℜπ−3/2−ivζ1+2ivΓ3/2+iv2iv+1/2dv=ζ1/2π3/428Γ3/4+π1/42Γ3/4−1/2.This result (compared with ([7], page 204, Lemma β)) can be verified numerically. The need to specify that only the real part of the integrand in (42) is to be used is twofold:(i)
The right-hand side is real and so must be the left-hand side(s).(ii)
Ostensibly, the integrand of (42) appears to be singular at v=0 unless one notes that(43)ζ1±2iv≈∓i2v+γ+⋯,where γ is Euler’s constant. Therefore, the real part of the integrand converges at v=0 and the imaginary part, singular at the origin, integrates to zero by antisymmetry.In general, whens=1/2+it, the left-hand side of both (38) and (39) are real, and so the equivalent of (42) applies. Written in terms of t, (42) generalizes to(44)ξ12+it=−12π3/2∫−∞∞ℜπ−ivζ1+2iv4t2+4iv+1Γ3/2+ivt2+2iv−4v2+1/4dv−12+π1/42Γ3/4.


## 5.1.1. The Caseσ=1/2

The result from (38) and (39) after setting η=0 (see Section 5.1) when s=1/2 is(42)12π∫−∞∞ℑξ1+2iv2iv+1/2vdv=∫−∞∞ℜπ−3/2−ivζ1+2ivΓ3/2+iv2iv+1/2dv=ζ1/2π3/428Γ3/4+π1/42Γ3/4−1/2.This result (compared with ([7], page 204, Lemma β)) can be verified numerically. The need to specify that only the real part of the integrand in (42) is to be used is twofold:(i)
The right-hand side is real and so must be the left-hand side(s).(ii)
Ostensibly, the integrand of (42) appears to be singular at v=0 unless one notes that(43)ζ1±2iv≈∓i2v+γ+⋯,where γ is Euler’s constant. Therefore, the real part of the integrand converges at v=0 and the imaginary part, singular at the origin, integrates to zero by antisymmetry.In general, whens=1/2+it, the left-hand side of both (38) and (39) are real, and so the equivalent of (42) applies. Written in terms of t, (42) generalizes to(44)ξ12+it=−12π3/2∫−∞∞ℜπ−ivζ1+2iv4t2+4iv+1Γ3/2+ivt2+2iv−4v2+1/4dv−12+π1/42Γ3/4.


## 5.2. The Casec = −5/4

In regions III and V, writing (35) in full gives(45)πc∫−∞∞π−ivζ−2c+2iv−2Γ−c+iv2iv−2s2−2c+2s−3−2c+2iv+s−3−2c+2iv−s−2dv=π1/42Γ3/4−1,or, equivalently(46)∫−∞∞2iv−2s2−2c+2s−3ξ−2c+2iv−2−2c+2iv+s−3−2c+2iv−s−2−2c+2iv−3dv=π1/42Γ3/4−1.Settingc=−5/4 in (45) leads to(47)∫−∞∞ζ1/2+2ivπ−5/4−ivΓ5/4+iviv−2s−12/42s−12/8+2v2dv=1−π1/42Γ3/4,a result that is valid for all s (see the dotted line in Figure 1(b)). Since the left-hand side is a function of s and the right-hand side is not, it must be true that the derivative of the left-hand side with respect to s vanishes. Performing this calculation (the integral is convergent, so the derivative and integral operator can be interchanged) yields(48)s−1/2∫−∞∞π−5/4−iv−2−iv+2s−12/4ζ1/2+2ivΓ5/4+iv2s−12/8+2v22dv=0.Compare (47) with Patkowski ([4], Theorem 1).


### 5.2.1.σ=1/2

Consider (47) in the case s=1/2, which gives(49)∫−∞∞iπ−5/4−iv2vζ12+2ivΓ54+ivdv=1−π1/42Γ3/4,and the integrand apparently has a singularity at v=0. However, a simple expansion of the integrand about that point shows that(50)ℜiπ−5/4−iv2vζ12+2ivΓ54+iv=−ζ1/22π1/4Γ3/4+Ov2,demonstrating that the real part of the integrand in (49) is nonsingular at v=0. However, near v=0, it is similarly shown that the imaginary part of the integrand diverges like v−1 and the imaginary part of the integral vanishes by antisymmetry about v=0.In the cases=1/2+it, (47) can be written in terms of ξ1/2+2iv as(51)1π∫−∞∞ξ1/2+2iviv+t2−t2/2+2v2−1/2+2ivdv=1−π1/42Γ3/4,whose integrand appears to become singular at v=±t/2. Writing the integrand in terms of its real and imaginary parts, and noting that ξ1/2+2iv is real, we find, for the real part(52)4π∫−∞∞ξ1/2+2iv16v2+1dv=1−π1/42Γ3/4,and, for the imaginary part,(53)∫−∞∞vξ1/2+2iv4v2−t216v2+1dv=0.Notice that in the former case, there is not dependence, so in (51), the variable t is in reality a free parameter, consistent with the argument applied to obtain (48). In the latter case, although a singularity exists, the integrand is antisymmetric about v=0, and so the singularities at ±v cancel.


## 5.2.1.σ=1/2

Consider (47) in the case s=1/2, which gives(49)∫−∞∞iπ−5/4−iv2vζ12+2ivΓ54+ivdv=1−π1/42Γ3/4,and the integrand apparently has a singularity at v=0. However, a simple expansion of the integrand about that point shows that(50)ℜiπ−5/4−iv2vζ12+2ivΓ54+iv=−ζ1/22π1/4Γ3/4+Ov2,demonstrating that the real part of the integrand in (49) is nonsingular at v=0. However, near v=0, it is similarly shown that the imaginary part of the integrand diverges like v−1 and the imaginary part of the integral vanishes by antisymmetry about v=0.In the cases=1/2+it, (47) can be written in terms of ξ1/2+2iv as(51)1π∫−∞∞ξ1/2+2iviv+t2−t2/2+2v2−1/2+2ivdv=1−π1/42Γ3/4,whose integrand appears to become singular at v=±t/2. Writing the integrand in terms of its real and imaginary parts, and noting that ξ1/2+2iv is real, we find, for the real part(52)4π∫−∞∞ξ1/2+2iv16v2+1dv=1−π1/42Γ3/4,and, for the imaginary part,(53)∫−∞∞vξ1/2+2iv4v2−t216v2+1dv=0.Notice that in the former case, there is not dependence, so in (51), the variable t is in reality a free parameter, consistent with the argument applied to obtain (48). In the latter case, although a singularity exists, the integrand is antisymmetric about v=0, and so the singularities at ±v cancel.


## 6. An Asymptotic Approximation toζσ+it

### 6.1. Integral Equations for the Real and Imaginary Parts ofξs

Consider (37)—region IV—in the general case s=σ+it, written in terms of the transfer function Mc,s=σ+it,v. The case c=−1 is of interest since the integral will span the 0-line, which connects to the 1-line by reflection (see (8)), and, acting in its capacity as a master function, ζ1±it has many well-known properties (e.g., [16]), the most relevant one being that it has no complex zeros. In addition, according to Figure 1(b), (37) is valid for all values of σ spanning the critical strip 0<σ<1. In this case, the transfer function Mx−1,σ+it,v and its real and imaginary components, shortened, except where necessary, to MRt,v and MIt,v, are, respectively,(54)Mx−1,σ+it,v=−2σ+2it−1σ+it2iv−1+σ+it2iv−σ−it+2iv−σ−it−1,(55)MRt,v=−8v2t2−σ2+σ+2tv1−2σ+2σ32−σ−4t2+3σ2+4t2+1σ−2t4−t2t2+4vt+σ2+4v2−2σ+1t2−4vt+σ2+4v2,(56)MIt,v=−8v3+−16σ+8tv2+−6t2+6σ2−6σ+2v+2σ−1tt2+4vt+σ2+4v2−2σ+1t2−4vt+σ2+4v2.Furthermore, these functions possess the following symmetry:(57)MRσ+it,v=MR1−σ+it,−v,(58)MIσ+it,v=−MI1−σ+it,−v,which symmetry also holds true for all values of c. A complete description of these functions is given in Appendixes A and B. From (37), the basic equation in region IV using c=−1 is(59)1−π1/42Γ3/4−ξσ+it=−iπ∫−∞∞MRt,v+iMIt,vϒ2ivvdv.It is now convenient to rewrite (59) as an integral over the range 0⋯∞ by first splitting the integral, setting v⟶−v and combining the two halves. The expression obtained is straightforward, although rather lengthy. A second lengthy equation can be obtained by first replacing σ⟶1−σ in (59) and similarly reducing the range to 0⋯∞. After splitting ϒ2iv into its real and imaginary components, these two equations can be added and subtracted, and with the help of (57) and (58), two new fundamental equations emerge:(60)π1/42Γ3/4−1=−ξRσ+it+1π∫0∞T1s,vϒR2iv+T2s,v−4ϒI2ivvdv,(61)ξIs=1π∫0∞T3s,vϒR2iv+T4s,vϒI2ivvdv,where(62)T1s,v≡−MIs,v+MIs,−v,(63)T2s,v≡−MRs,v−MRs,−v+4,(64)T3s,v≡MRs,v−MRs,−v,(65)T4s,v≡−MIs,v−MIs,−v.In this manner, becauseξs is self-conjugate, the real and imaginary components of ξs are isolated, and expressed in terms of convergent integrals. Furthermore, subtracting (D.2) from (60) yields a simpler form of (60):(66)ξRs=1π∫0∞T1s,vϒR2iv+T2s,vϒI2ivvdv.For the caset=0,0<σ<1, (66) reduces to a relation between ξσ on a section of the real line, and ϒ2iv on the complex line ℜv=0, specifically(67)ξRσ=1π∫0∞1σ2+4v2σ−12+4v24v23σσ−1+4v2+1ϒR2iv+v64v4+16σ2−σ+1v2−2σσ−1ϒI2ivvdv.


### 6.2. The Background Terms

Although the functionsT1,2,3,4s,v look formidably complicated when written in full, each can be written in a more transparent form when decomposed using partial fractions. Specifically, with c=−1,(68)T1s,v=2σv−tt−2v2+σ−12+21−σv−tt−2v2+σ2+2σv+tt+2v2+σ−12+21−σv+tt+2v2+σ2,(69)T2s,v=−tt−2v+1−σσt−2v2+σ2+−tt−2v+1−σσt−2v2+1−σ2+−tt+2v+1−σσt+2v2+σ2+−tt+2v+1−σσt+2v2+1−σ2+4,(70)T3s,v=−tt−2v−σσ−1t−2v2+σ−12+tt−2v+σσ−1t−2v2+σ2+tt+2v+σσ−1t+2v2+σ−12+−tt+2v−σσ−1t+2v2+σ2,(71)T4s,v=−2vσ−1−tt−2v2+σ2+−2σv+tt−2v2+σ−12+2σv+tt+2v2+σ−12+2vσ−1−tt+2v2+σ2.Remark.
Although not evident when written in the decomposed form, when expressed in factored form, both (70) and (71) contain an overall factor σ−1/2, a requirement that ξIs vanishes when σ=1/2. Furthermore, it is easily seen that T1,2σ+it,v=T1,21−σ+it,v, T3,4σ+it,v=−T3,41−σ+it,v, T1,2σ+it,v=T1,2σ−it,v, and T3,4σ+it,v=−T3,4σ−it,v.
As written, the first two terms of each of the above explicitly demonstrate the existence of a pole in the complexv-plane at v=±t/2±iσ, along with invariance under the symmetry σ⟶1−σ. Since we are in general interested in the case t⋙σ,1−σ>0, such poles lie relatively close to the positive real v-axis as demonstrated by the nature of the peak(s) in Figure 14 (see Appendix A). By reducing the integral to the range 0⋯∞, the influence of poles corresponding to negative values of v has been effectively removed from the integrand v>0. Drawing on these observations, it is suggestive that the functions T1,2,3,4s,v can be separated into two components—pole terms associated with the first two terms of each, plus background terms associated with all four terms, in each of the above; it is further suggestive that only the pole terms will contribute to the integrals (see Figures 16 and 18), thereby allowing an approximate solution to be obtained. How can this be done?
Consider (66). From (68) and (69), it would appear superficially that both T1s,v and T2s,v vanish as 1/t for large values of t, which immediately demonstrates a potential inconsistency—the left-hand side of (66) vanishes exponentially and so must the right-hand side. Since the remaining integrand factors ϒR2iv and ϒI2iv contain no t dependence, the only possibility is that 1/t dependence must vanish from the (highly oscillatory) integral, and somewhere buried in the higher order asymptotic terms will be found a term with exponentially decreasing t dependence. Furthermore, the possibility also exists that cancellations will occur between the two integrand terms containing T1s,v and T2s,v in (66). Any attempt at a numerical evaluation of (66) for reasonably large values of t will immediately demonstrate the truth of this prediction in the form of a severe cancellation of significant digits. An example follows.
Chooseσ=1/3 and t=50, chosen sufficiently large to distinguish inverse powers of t from exponentially decreasing terms of the form exp−bt, but not so large that multidigit computer arithmetic must be used. For this case, calculated with 15 digits, we find(72)1π∫0∞vT113+50i,vϒR2ivdv=0.000810442386190651,(73)1π∫0∞vT213+50i,vϒI2ivdv=−0.0008104423861872670,and the sum 3.3840×10−15 is comparable with ξ1/3+50i=3.38361×10−15. Clearly, each of the integrals has an absolute value much greater than the sum of the two, and it is only because of a cancellation of digits that the final result can be found. Here, we see that the cancellation of digits has resulted in the loss of 10 digits from the sum; for larger values of t, the effect will be more significant, and one might despair of utilizing (66) for anything. It will now be shown that these (and other) cancellations can be analysed analytically, and an exponentially decreasing term can be extracted, which is the basis for the approximate model solution.
First of all, forv≤t/2, consider the asymptotic expansion t⟶∞ of each of the individual terms composing T1 labelled, respectively, by a second subscript:(74)T1,1s,v⟶−1t+2σv−4vt2+42σ−3v2+σ−12t3+Ot−4,(75)T1,2s,v⟶−1t+21−σv−4vt2+−42σ+1v2+σ2t3+Ot−4,(76)T1,3s,v⟶1t+21−σv−4vt2+42σ+1v2−σ2t3+Ot−4,(77)T1,4s,v⟶1t+2σv−4vt2+−42σ−3v2−σ−12t3+Ot−4.
Clearly, the leading asymptotic dependence1/t cancels between all of “pole” and “background” terms in pairs, demonstrating that the pole terms also contribute significantly to the asymptotic background. This verifies that a numerical evaluation of the T1s,v integral will be challenging as t increases. More importantly, it is apparent that the overall asymptotic dependence of the T1s,v part of the integral will have the leading dependence 1/t2 because the multiplicative factor ϒR2iv lacks t dependence as noted. But, as also noted, this is inconsistent with (66), where the left-hand side has an exponentially decreasing asymptotic behaviour in t. One explanation for this inconsistency could be that there exists a cancellation between the two terms in the integrand labelled by T1 and T2 to at least order 1/t2. This would explain the cancellation observed in the numerical results (72) and (73). The following gives the asymptotic expansion of each of the corresponding elements of T2s,v in analogy to (74)–(77):(78)T2,1s,v⟶−1−2vt+−4v2+σt2+−8v3+2σσ+2vt3+Ot−4,(79)T2,2s,v⟶−1−2vt+−4v2−σ+1t2+−8v3+2σ2−8σ+6vt3+Ot−4,(80)T2,3s,v⟶−1+2vt+−4v2+σt2+8v3−2σσ+2vt3+Ot−4,(81)T2,4s,v⟶−1+2vt+−4v2−σ+1t2+8v3+−2σ2+8σ−6vt3+Ot−4.
Again, we see a cancellation between each of the four terms to order1/t as well as a cancellation of the terms of order t0 with the fifth term in (69). And, again we find an inconsistency with (66) because the leading term of T2s,v is also of order 1/t2. Define each term corresponding to t−k in the asymptotic series of T1s,v by(82)T1˜k≡coefficient  oft−2kin the serieslimt⟶∞v<tT1s,v,and similarly for T2s,v. When the first two asymptotic terms of the factors in the integrand on the right-hand side of (66) are written in full, we find(83)T1s,v∼−12vt2+4v9σ2−20v2−9σ+4t4+Ot−6,(84)T2s,v∼−16v2+2t2+−64v4+48σ2−48σ+48v2−6σ2+6σ−2t4+Ot−6,so that(85)T˜11=−12vt2,(86)T˜12=4v9σ2−20v2−9σ+4t4,(87)T˜21=−16v2+2t2,(88)T˜22=−64v4+48σ2−48σ+48v2−6σ2+6σ−2t4,and clearly there can be no cancellation of terms of equal order in 1/t2, or 1/t4 between terms containing T1s,v and T2s,v. The only remaining possibility is that the integral containing each of these asymptotic forms itself vanishes identically, at least to the orders discussed above. In fact, this must be true to all asymptotic orders in t−2k,k≥0 because it is impossible for a series of the form ∑k=1∞akt−2k to add up to an exponential of the form exp−bt. Furthermore, if the integrals vanish as predicted, they must vanish for all σ. Otherwise, (66), which is an exact result cannot be asymptotically and universally true. And somewhere, accompanying all these dominant terms that vanish asymptotically to inverse polynomial order, exist other terms associated with the pole terms, that carry the exponentially vanishing result corresponding to the left hand-side of (66) in the asymptotic limit t⟶∞. Numerically, these lesser terms equate to the sum of (72) and (73). It will now be proven that the integrals do in fact vanish, as predicted, to at least the first few orders in t−2k,k=1,2,3, following which a model will be constructed to extract the terms that instead vanish exponentially.
Based on the first terms of (83) and (84), it is predicted that the sum of the integrals involving t−2 must vanish, that is,(89)1t2∫0∞−12v2ϒR2iv+v−16v2+2ϒI2ivdv=0.Proof.
Consider the classical result ([7], equation (2.15.6)) relating the inverse Mellin transform of ϒc+iv to the Jacobi Θ3 function:(90)∫−∞∞ϒc+ivdv=4π∑n=1∞e−n2x=2πΘ30,exp−πx−1,valid for c>1. We are interested in the case c=0, so shift the contour by subtracting the residue at v=1−c/i and half the residue at v=ic resulting in the identity(91)π2∑n=1∞e−n2x−πx+12=∫0∞πivx−iv+π−ivxivϒR2ivdv+i∫0∞πivx−iv−π−ivxivϒI2ivdv,if c=0, after converting the integration limits to 0,∞ and writing ϒ2iv as the sum of its real and imaginary parts. By an increasing sequence of higher order derivatives, evaluated at x=π, it is possible to obtain relevant sums using identities given by Romik [13]. The first seven such sums are listed in Appendix D, from which it is possible to obtain the first seven even moments of ϒR2iv and odd moments of ϒI2iv, listed also in Appendix D. Substituting the moments (D.2)–(D.4) into (89) verifies that the prediction (89) is true. QED.
Employing similar logic, a further prediction arises regarding the next term of order1/t4. It too must vanish for all σ. From (83), (84) and (89) we predict(92)1t4∫0∞ϒR2ivv4+v96v4−40v2−1ϒI2iv120dv=0.
Note thatσ dependence has disappeared from (92), as expected. By substituting (D.2)–(D.6) into (92) the truth of this prediction can be verified. This sequence of forecasts can be continued to the next recursive level, producing the forecast identity:(93)1t6∫0∞224ϒR2ivv6−−128v7+112v5+28v33+v3ϒI2ivdv=0.
Again,σ dependence has cancelled of its own volition. The truth of the prediction (93) can similarly be verified by substituting the set of moments (D.2)–(D.8) into (93).
To summarize, we have the following proposition.Proposition 1 (deduced and partially demonstrated).
The above sequence of predictions can be proven in general for all terms on the right-hand side of (66) that asymptotically vanish to inverse polynomial order t−2k,k>3 independently of σ.Whatever remains must then be of exponentially decreasing order and must therefore equate to the left-hand side of (66), and the challenge is to somehow isolate such term(s). The cancellation of integrals of asymptotically vanishing inverse polynomial order explains the loss of significant digits observed between (72) and (73) and suggests that such loss of significant digits will only increase for larger values of t, as the interested reader can verify for herself.Expressed another way, the crucial point is that although the peaks shown in Figure14 originate from the pole terms, these pole terms also contribute to the background terms of inverse polynomial order t−2k,k>1 that cancel. Nonetheless, the signal associated with the pole terms naturally separates from the background. The separation of signal from background is the essence of the approximate solution to be presented here.To gain further insight into these issues, consider Figure2(a) which shows (the absolute value of) the integrand:(94)J1s=σ+it,v≡T1s,vϒR2iv,as a function of v for several values of t at σ=1/3. Similarly, Figure 2(b) shows the same result for the second term in the integrand defined by(95)J2s=σ+it,v≡T2s,vϒI2iv.Three properties are immediately evident: most of the absolute value of both integrands occurs nearv=0, both J1s,v and J2s,v decrease exponentially with increasing v and are of similar magnitude—a necessity if they are to cancel numerically (see (72) and (73)). A fourth significant property is that the pole term, embedded in T1s,v at v=t/2 is clearly visible above the exponentially decreasing background in each curve of Figure 2(a), although it lies many orders of magnitude below the numerically significant portion of the integrand. In fact, it appears to be enhanced as t increases, as might be expected because the magnitude of its peak varies as t—see (C.1)—whereas the background varies as t−2k,k>1. However, in Figure 2(b) the zero of T2s,v is not readily apparent because of the resolution of each curve; moreover, each of the curves in Figure 2(b) is consistent with what appears to be the background as discussed previously. It will now be shown that these observations can be used to extract a signal from the background, and in so doing obtain an approximate solution to (66).Figure 2
The individual terms composing the integrand of (66) as a function of the integration variable v, using σ=1/3 for several values of t. Each curve has been scaled by a (smooth) factor expπt/4/t2 to separate them for easier viewing; otherwise, all of the curves coincide to a good degree of approximation, except near v≈0. (a) An illustration of the integrand J11/3+it,v at several values of t=12.5,25,50,100,200; the arrows mark the points v = t/2 that being the location of the peak of the pole term contained in T1s,v for each of the curves at each individual value of t. (b) An illustration of the integrand J21/3+it,v at several values of t=12.5,25,50,100,200; the arrows mark the points v = t/2 that being the location of the zero of the pole term contained in T2s,v for each of the curves at each individual value of t.
(a)
(b)


### 6.3. The Signal

Consider Figure3(a) which illustrates the peak of the function T1s,v over a wide range of v using s=1/3+104i. Also shown are the cumulative background terms ∑k=1NT˜1k. It can be seen that as N increases, the sum of the background terms give a better and better approximation to the background part of the function T1s,v when v<t/2. In fact, when the each of the cumulative background terms are subtracted from T1s,v, that part of the function associated with small values of v decreases (Figure 3(b)) as the number of background terms increases. But it is precisely small values of v that contribute most to the integral itself (see Figure 2). And as we have seen from (89), (92) and (93), when coupled with the full part of the integrand associated with T2s,v the integral associated with each of these background terms has either been proven, or predicted, to vanish. What is left? Only the pole term remains, wherein resides the sought after “signal.”Figure 3
The functionT11/3+104i,v in comparison with the first few cumulative background terms ∑k=1NT˜1k for wide range of v near t/2. In both figures, the vertical arrow marks the location of the peak at v=t/2. (a) An illustration of the function T11/3+104i,v as well as the cumulative first few background terms ∑k=1NT˜1k for the first 5 values of N for a wide range of v about t/2. (b) An illustration of the function T11/3+104i,v as well as the effect of subtracting the first few cumulative background terms ∑k=1NT˜1k for the first 5 values of N for a wide range of v about t/2.
(a)
(b)Remark.
Because absolute values have been used, the corresponding Figures for the functionT2s,v are qualitatively identical to Figure 3.
To understand how all this affects the integral itself, consider Figure4 which shows both integrands J1s,v and J2s,v, again using s=1/3+it with t=104, in close proximity to the peak region v=t/2. Since a logarithmic scale is used, in both figures, the alternating function(s) J12s,v are shown (dotted) colour coded, positive (red), and negative (blue). The green (dash-dot) line shows the leading asymptotic behaviour of J12s,v originating from the factor Γiv embedded in the factor ϒ2iv, scaled to fit the Figure. This is the background.
Focusing on Figure4(a), the peak of the factor T1s,v is indicated by a magenta (dashed) line, also marked by the vertical arrow. The effect of the peak of T1s,v on the integrand J1s,v can be seen as an enhancement of the integrand in the immediate vicinity of the peak. This is indicated by the superimposed solid curve representing J1s,v, again colour coded red and blue, but only for those values of t/2−2<v<t/2+2. The top horizontal dashed line equates to the ordinate of the point J1s,t/2 and the lower line delineates points lying one order of magnitude smaller that the ordinates at the peak. Figure 4(b) is similar to Figure 4(a), with the exception that instead of a peak at v=t/2, the curve representing J2s,v vanishes (see Appendix C).Figure 4
J
1
s
,
v and J2s,v in close proximity to the peak of T1s,v or zero of T2s,v, respectively, at v=t/2, as well as the shape of the background of both. (a) An illustration of the function J11/3+104i,v accompanied by T11/3+104i,v and the scaled asymptotic shape of ℜϒ11/3+104i,v very close to v = t/2. (b) An illustration of the function J21/3+104i,v accompanied by T21/3+104i,v and the scaled asymptotic shape of ℜϒ11/3+104i,v very close to v = t/2.
(a)
(b)


### 6.4. The Approximate Solution

Figures4(a) and 4(b) hold the key to the approximate solution of (66). Up to this point, all analysis has consisted of observation, (generalized) deduction, and a small amount of proof. The crucial point to be extracted from Figure 4(a) is that the peak of T1s,v “extrudes” the integrand of J1s,v above background in a small region about v=t/2. Although the background (dotted part of the curve) represents the vast majority of the magnitude of the integrand over many orders of magnitude, when this background is decomposed into terms that are of inverse polynomial order (asymptotically in t), the integration of these terms cancels between the two terms labelled by T1s,v and T2s,v and illustrated in Figures 4(a) and 4(b) (see Section 6.2), and only that part of the curve near v=t/2 (shown solid) contributes to the integral. Conversely, with reference to Figure 4(b), the strong zero of the function T2s,v acts to depress the function J2s,v near v=t/2, although the background terms still assume their role of acting to cancel similar background terms shown in Figure 4(a). All these suggest an approximation that should give a reasonable representation of a solution to (66).


#### 6.4.1. Approximation 1

Let(96)Tjs,v=T˜js,v+Tojs,v,where T˜js,v represents the background, Tojs,v represents the nonbackground associated with the pole and j=1,2. From (82), we have(97)T˜js,v=∑k=1∞T˜jk.When applied to (66), this leads to(98)Js,v≡J1s,v+J2s,v=T˜1s,v+To1s,vϒRs,v+T˜2s,v+To2s,vϒIs,v=T˜1s,vϒRs,v+T˜2s,vϒIs,v+To1s,vϒRs,v+To2s,vϒIs,v.But, according to Section6.2 (i.e., Proposition 1),(99)∫0∞T˜1s,vϒRs,v+T˜2s,vϒIs,vvdv=0;thus, (66) becomes(100)ξRs=1π∫0∞T1os,vϒRs,vvdv+1π∫0∞T2os,vϒIs,vvdv.SinceTjos,v represents the nonbackground (pole) terms this leads to Approximation 2.


#### 6.4.2. Approximation 2

Let(101)ξRs=1π∫t/2−δ1σ,tt/2+δ1σ,tT1os,vϒRs,vvdv+1π∫t/2−δ2σ,tt/2+δ2σ,tT2os,vϒIs,vvdv,where δjσ,t is a (small) function of Oσ0,Ot0 associated with each of Tojs,v—see (C.4); that is, only that part of the integrand in close proximity to v=t/2 contributes to the overall value of the integral, which integral is approximately symmetric about the point v=t/2. Everything else cancels. Equivalently, this recognizes that only that section of the curve coloured solid in Figures 4(a) and 4(b) will contribute to the integral, notwithstanding the fact that the dotted sections ostensibly contribute to the integrand by many more orders of magnitude. This is demonstrated numerically below (see (105)).Remark.
For clarity, I writeδjσ,t≡δj, recognizing that δjσ,t is a simple and low-order function of both σ and t; the functional dependence is investigated later.


#### 6.4.3. Approximation 3

Since each of theδj are much less than any value of t in any range of interest, first rewrite (101) as(102)ξRs=δ12π∫−11T1os,t2+δ1yϒRi2δ1y+t2δ1y+tdy+δ22π∫−11T2os,t2+δ2yϒIi2δ2y+t2δ2y+tdy.By applying an obvious change of variables, expand (102) about δj=0, and consider only those terms of leading asymptotic order in t, giving the following to first order in δj and asymptotic t:(103)ξRs≈t2δ1ϒRitσσ−1π+tδ2σ2−σ−1ϒIitσσ−1π.Notice that the second term in (103) is smaller than the first term by one power of t, reflecting the fact that To2s,v vanishes at v=t/2. This means that asymptotically, we expect the second term in (103) to vanish, relative to the first term, except if ϒRit≈0.Remark.
By expanding aboutδj=0, we are selecting for values of the integrand at v=t/2. This could be interpreted as the introduction of a parameter w1 to model the width of T1os,v near its peak—effectively replacing T1os,v by inserting a box of equal area around the peak and thereby approximating parts of the integrand near v=t/2 which dominate T1os,v and thereby, the integral itself. According to the Mean Value Theorem for integrals, this can always be done and w1≠0. The parameter w1 is generalized by δ1 which captures the area of the product term T1os,vϒRiv and reasonable models for this parameter should be considered in future studies, recognizing that it is a (weak) function of both σ and t (but see Section 8). Similarly, since T2os,v vanishes antisymmetrically about v=t/2, it is reasonable to expect that it will integrate somewhat close to zero. Thus, we expect that terms involving δ2 will be much smaller than those involving δ1 which appears to be the case for large values of t. Here, let δ1=σ1−σ/2 based on numerical experimentation with the above approximation, and later, δ4=δ1. Since the relative width of the peak of T1os,v decreases for large values of t, this approximation should improve as t increases (see Figure 16).
With reference to the example presented in (72) and (73), the two terms on the right-hand side of (103) with δ1=δ2=0.2357 (=1/32) give(104)ξR13+50i≈3.63105×10−15−1.01201×10−16=3.52985×10−15,compared with the correct result ξR1/3+50i=3.3836×10−15, a reasonable approximation. Furthermore, setting δ11/3,50=δ21/3,50=0.2357 in (101) gives(105)1π∫24.7642925.2357T1os,vϒRs,vvdv+1π∫24.7642925.2357T2os,vϒIs,vvdv=2.3675×10−15+5.0984×10−16=2.8773×10−15.where I have used T1,2os,v=T1,2s,v since the latter is dominated by the pole term over the narrow range of integration (see Figure 18(a)). Note that the second term associated with δ2 is numerically smaller than the term associated with δ1 as predicted. Significantly, the order of magnitude of (105) is correct, demonstrating that the signal has been extracted, although the numerical estimate is not as good as the approximate one because the functional dependence of δ1,2 has not been generally selected for this purpose. In fact, resetting the limits of (105) using δ1=δ2=0.2983 yields the correct result to four significant figures. Of course, there is no reason to expect that δ2=δ1; this becomes less of an issue as t increases since the relative contribution from the T2os,v term asymptotically vanishes except at a zero.


#### 6.4.4. Approximation 4

To obtain an asymptotic approximation ofζRσ+it as t⟶∞, first write the left-hand side of (103) in polar form and employ (7), to obtain(106)ξRs⟶−e−πt/4π−σ/2ζσ+it2πt/2σ/2−1/2σ−12tsinϕ+t2cosϕ2.Now, consider the right-hand side of (103) by identifying ϒRit in terms of its components (see (3)), apply the same approximations employed above, reunite both sides of the equation, cancelling common nonzero factors on both sides and eventually arrive at(107)ζsσ−12sinϕ+t2cosϕ≈tcosΦδ1+σ2−σ−1sinΦδ2/tπσ1−σζitt2π−σ/2.Starting from (61), a similar set of approximations can be applied. As in the model for ξRs and with reference to Figure 18, the background terms obtained from the decomposition of functions T3s,v and T4s,v integrate to zero exactly as in Section 6.2 with the change that each term contains an additional overall multiplicative factor of 2σ−1/t. Again, the dominant contribution to ξIs arises from the pole term associated with T4s,v. Thus, the equivalent of (103) is(108)ξIs≈22σ−1t2δ4ϒIit+δ3/tϒRitπσ1−σ,and the equivalent of (106) is(109)ξIs⟶e−πt/4π−σ/2ζσ+it2πt2σ/2−1/2σ−12tcosϕ−t22sinϕ.Again, because0<σ<1≪t, follow the exact steps used previously and ultimately arrive at the equivalent of (107)(110)ζs12−σcosϕ+tsinϕ≈4t1/2−σsinΦδ4+cosΦδ3/tπσ1−σζitt2π−σ/2.Remark.
Ifσ=1/2 in (110), sinϕ on the left-hand side vanishes (see (9) and Section 6.5).
Furthermore, it is possible to extract approximations forζRσ+it and ζIσ+it by solving (107) and (110) simultaneously, using a simple trigonometric expansion because argζσ+it is embedded within the definition of ϕ (see (13)), by noting that cosargζsζs=ζRs (and similarly for ζIs). Asymptotically (because both cosΦ and sinΦ cannot both vanish simultaneously), this solution yields a simple representation after applying the next approximation.


#### 6.4.5. Approximation 5.

Let(111)δ2t=δ3t=0,giving(112)ζRσ+it≈2ζit−cosωcosΦδ1−2σ−1/2sinΦsinωδ4σσ−1πt2π−σ/2,(113)ζIσ+it≈2ζit−sinωcosΦδ1+2σ−1/2cosωsinΦδ4σσ−1πt2π−σ/2,where(114)ω=argΓσ2+it2−t2logπ.Notice that the right-hand sides of both (112) and (113) do not contain any components of ζσ+it, and therefore the left-hand sides are expressed in terms of independent functions (compare with [17]).


#### 6.4.6. Numerical Examples

Figure5 illustrates both approximations (112) and (113) for an arbitrary choice of σ. From (103), it is seen that any structure in ξRσ+it must originate from the function ϒit on the right-hand side, and this function lacks σ dependence. Thus, (103) predicts that the structure of ξRσ+it, particularly the location of the zeros, must be relatively independent of σ since, as was noted in Section 6.1, σ dependence only resides in the transfer functions and to a lesser extent any model developed to approximate δ1,2 as a function of σ and/or t. Thus, in (66), if ϒRiv happens to vanish at v=t/2, then the peak associated with T1t,v is multiplied by a vanishing factor and contributions to the integral from both functions T1t,v=t/2 and T2t,v=t/2 vanish, independently of σ. So, the location of the zeros is approximately governed by cosΦ=0 (see the discussion surrounding (124) in Section 7). This prediction can be checked (see Figure 6(a)) which demonstrates that this property appears to be approximately true, for two randomly chosen, disparate values of σ.Figure 5
The approximations (112) and (113) to ζRσ+it and ζIσ+it, respectively, near t=500 using δ1=δ4=σ1−σ/2. (a) A comparison of the left- and right-hand sides of (112) for σ = 0.7 near t = 500. (b) A comparison of the left- and right-hand sides of (113) for σ = 0.7 near t = 500.
(a)
(b)Figure 6
For several values ofσ and t, this is a comparison of ξRσ1+it and ξRσ2+it as they exist (left) and as they are predicted to exist by the approximation (right) (see (118)). In both cases, the functions have been scaled by a factor expπt/4/t2. (a) A comparison of ξRσ+it for two very different values of σ near t = 700. (b) A comparison of the left- and right-hand sides of (118) for two very different values of σ near t = 500.
(a)
(b)Figure 7
A simple model for bothδ1σ,t and δ4σ,t exists as a function of σ, although the dependence on t may not be simple (see Figure 10(a)). (a) The solution function Δ1σ,t for several values of t over a range of σ. Each of the curves has been normalized by Δ11/2,t with values shown. (b) The solution function Δ4σ,t for several values of t over a range of σ. Each of the curves has been normalized by Δ41/2,t with values shown.
(a)
(b)From this observation, arises a constraint on the approximation. Consider two instances of (103) with different choices σ=σ1 and σ=σ2. The approximation predicts that(115)ξRσ1+itξRσ2+it=δ1σ1σ21−σ2δ1σ2σ11−σ1,since the model suggests that δ1 is at least a (weak) function of σ. Now, suppose that σ2=1−σ1, in which case the left-hand side of (115) becomes unity (see (2)); for consistency, this imposes a constraint on the parameter δ1, specifically(116)δ1σ=δ11−σ.With this constraint in mind, and consistent with the approximations specified previously, it is possible to obtain some insight into a reasonable model forδ1σ,t by solving (103) using the approximation δ2/t=0 (see Approximation 5). Introduce σ,t dependence by letting the solution be denoted by Δ1σ,t and consider Δ1σ,t over the range 0≤σ≤1, for several choices of t. The result is given in Figure 7(a), providing evidence that it is likely that a smooth function exists to capture the σ dependence of δ1σ,t, although the normalization factor varies considerably as a function of t. The somewhat arbitrary and simple model(117)δ1σ,t=σ1−σ2,used throughout this paper is also shown; it is clear that this model, although a reasonable first approximation, requires future refinement. A similar result can be found for Δ4σ,t, the solution of (108) (see Figure 7(b)). (Remark: to obtain the normalization of the latter figure, expand the function ξσ+it about the point σ=1/2 and cancel the numerator and denominator factors σ−1/2 that appear in the imaginary part).Figure 8
A comparison of the left- and right-hand sides of (107) for two very different values of σ near t=900. The parameter δ1=σ1−σ/2 in both cases, and all curves have been normalized by the factor 1/t.Figure 9
A comparison of the left- and right-hand sides of (119) near t=900. The parameter δ1=σ1−σ/2 and σ=1/2.All this suggests that it is of interest to investigateξσ+it for two different choices of σ. (115) and (117) predict that(118)σ11−σ1ξRσ1+it≈ξRσ2+itσ21−σ2,an example of which is shown in Figure 6(b). Finally, it is of interest to consider approximations to ζσ+it as given in (107) (see Figure 8).Figure 10
A test (a) of the solution to (121) over the range 700≤t≤720 applied (b) to that same equation over a completely different range of t. (a) The curve in this Figure shows the solution Δ41/2,t of (121) over a range of t. The horizontal line Δ41/2,t≈0.16 illustrates a first approximation. (b) This is a comparison of the absolute value of the right and left-hand sides of (120) using Δ41/2,t=0.16.
(a)
(b)


### 6.5. The Caseσ=1/2

From the real part ofξs, (107) is of particular interest when σ=1/2. By straight substitution, using (117) and (8), we obtain(119)ζ12+itcosϕ≈4δ112,tcosΦζ1−itt1/423/4π5/4.This is consistent with equation (5.1.8) in [7]; for a numerical example, see Figure 9. In (119), note that the factor cosϕ=−1k because of (13), where k approximates the number of zeros of ζ1/2+it relative to t=0. For further discussion, see Section 8.2.From the imaginary part ofξs, in the case of (108) with σ=1/2, both sides vanish, as they must because it is known that ξI1/2+it=0. This is also true of (110) because in the limit σ=1/2, we have ϕ=kπ due to (9) and (13). From (10), by straightforward differentiation of (2) applied to (108), we obtain(120)limσ⟶1/2ξI1/2+itσ−1/2=ℑ∂∂σξσ+itσ=1/2=−1kΓ1/4+it/2π1/4t−14t2+14ℑψ14+it2ζ12+it+12t2+14sinα−βζ′12+it≈4t2δ4ϒIit+δ3/tϒRitπσ1−σ,where ψs is the digamma function and α and β are defined in (16) and (17), respectively. Asymptotically this becomes(121)2π3/412ψI14+it2ζ12+it−sinα−βζ′12+it≈−64δ4ζitsinΦt−1/4.Figure 11
A comparison of the predicted relationship betweenξRσ+it (left) and ξIσ+it (right) versus the corresponding real and imaginary components of ϒit at two different choices of σ near t=800. The figures are scaled by eπt/4/t2. (a) A comparison of the left- and right-hand sides of (103) near t = 800. The parameter δ1 =σ1−σ/2, σ=1/2 and δ2=0. (b) A comparison of the left- and right-hand sides of (108) near t = 800. The parameter δ4 =σ1−σ/2, σ=0.22 and δ3=0.
(a)
(b)For this fixed value ofσ=1/2, it is possible to obtain an estimate of δ41/2,t by solving (121) for δ4. Denote the solution by Δ41/2,t and consider Figure 10(a) which shows Δ41/2,t for a range of t. As a first approximation, choose δ41/2,t=0.16 as suggested in the figure. It should be noted that the denominator of the solution can vanish, but the numerator does not vanish simultaneously; these singular points of the solution are apparent in the figure; however, at such points, the parameter δ31/2,t must be incorporated into the model; this will prevent the singularity from occurring. Thus, the approximate solution in Figure 10(a) is valid in between the singular points; the horizontal line Δ41/2,t≈0.16 used in this example has been chosen on that basis. Figure 10(b) uses δ4 = .16 and δ3 = 0 to compare the right- and left-hand sides of (120).


## 6.1. Integral Equations for the Real and Imaginary Parts ofξs

Consider (37)—region IV—in the general case s=σ+it, written in terms of the transfer function Mc,s=σ+it,v. The case c=−1 is of interest since the integral will span the 0-line, which connects to the 1-line by reflection (see (8)), and, acting in its capacity as a master function, ζ1±it has many well-known properties (e.g., [16]), the most relevant one being that it has no complex zeros. In addition, according to Figure 1(b), (37) is valid for all values of σ spanning the critical strip 0<σ<1. In this case, the transfer function Mx−1,σ+it,v and its real and imaginary components, shortened, except where necessary, to MRt,v and MIt,v, are, respectively,(54)Mx−1,σ+it,v=−2σ+2it−1σ+it2iv−1+σ+it2iv−σ−it+2iv−σ−it−1,(55)MRt,v=−8v2t2−σ2+σ+2tv1−2σ+2σ32−σ−4t2+3σ2+4t2+1σ−2t4−t2t2+4vt+σ2+4v2−2σ+1t2−4vt+σ2+4v2,(56)MIt,v=−8v3+−16σ+8tv2+−6t2+6σ2−6σ+2v+2σ−1tt2+4vt+σ2+4v2−2σ+1t2−4vt+σ2+4v2.Furthermore, these functions possess the following symmetry:(57)MRσ+it,v=MR1−σ+it,−v,(58)MIσ+it,v=−MI1−σ+it,−v,which symmetry also holds true for all values of c. A complete description of these functions is given in Appendixes A and B. From (37), the basic equation in region IV using c=−1 is(59)1−π1/42Γ3/4−ξσ+it=−iπ∫−∞∞MRt,v+iMIt,vϒ2ivvdv.It is now convenient to rewrite (59) as an integral over the range 0⋯∞ by first splitting the integral, setting v⟶−v and combining the two halves. The expression obtained is straightforward, although rather lengthy. A second lengthy equation can be obtained by first replacing σ⟶1−σ in (59) and similarly reducing the range to 0⋯∞. After splitting ϒ2iv into its real and imaginary components, these two equations can be added and subtracted, and with the help of (57) and (58), two new fundamental equations emerge:(60)π1/42Γ3/4−1=−ξRσ+it+1π∫0∞T1s,vϒR2iv+T2s,v−4ϒI2ivvdv,(61)ξIs=1π∫0∞T3s,vϒR2iv+T4s,vϒI2ivvdv,where(62)T1s,v≡−MIs,v+MIs,−v,(63)T2s,v≡−MRs,v−MRs,−v+4,(64)T3s,v≡MRs,v−MRs,−v,(65)T4s,v≡−MIs,v−MIs,−v.In this manner, becauseξs is self-conjugate, the real and imaginary components of ξs are isolated, and expressed in terms of convergent integrals. Furthermore, subtracting (D.2) from (60) yields a simpler form of (60):(66)ξRs=1π∫0∞T1s,vϒR2iv+T2s,vϒI2ivvdv.For the caset=0,0<σ<1, (66) reduces to a relation between ξσ on a section of the real line, and ϒ2iv on the complex line ℜv=0, specifically(67)ξRσ=1π∫0∞1σ2+4v2σ−12+4v24v23σσ−1+4v2+1ϒR2iv+v64v4+16σ2−σ+1v2−2σσ−1ϒI2ivvdv.


## 6.2. The Background Terms

Although the functionsT1,2,3,4s,v look formidably complicated when written in full, each can be written in a more transparent form when decomposed using partial fractions. Specifically, with c=−1,(68)T1s,v=2σv−tt−2v2+σ−12+21−σv−tt−2v2+σ2+2σv+tt+2v2+σ−12+21−σv+tt+2v2+σ2,(69)T2s,v=−tt−2v+1−σσt−2v2+σ2+−tt−2v+1−σσt−2v2+1−σ2+−tt+2v+1−σσt+2v2+σ2+−tt+2v+1−σσt+2v2+1−σ2+4,(70)T3s,v=−tt−2v−σσ−1t−2v2+σ−12+tt−2v+σσ−1t−2v2+σ2+tt+2v+σσ−1t+2v2+σ−12+−tt+2v−σσ−1t+2v2+σ2,(71)T4s,v=−2vσ−1−tt−2v2+σ2+−2σv+tt−2v2+σ−12+2σv+tt+2v2+σ−12+2vσ−1−tt+2v2+σ2.Remark.
Although not evident when written in the decomposed form, when expressed in factored form, both (70) and (71) contain an overall factor σ−1/2, a requirement that ξIs vanishes when σ=1/2. Furthermore, it is easily seen that T1,2σ+it,v=T1,21−σ+it,v, T3,4σ+it,v=−T3,41−σ+it,v, T1,2σ+it,v=T1,2σ−it,v, and T3,4σ+it,v=−T3,4σ−it,v.
As written, the first two terms of each of the above explicitly demonstrate the existence of a pole in the complexv-plane at v=±t/2±iσ, along with invariance under the symmetry σ⟶1−σ. Since we are in general interested in the case t⋙σ,1−σ>0, such poles lie relatively close to the positive real v-axis as demonstrated by the nature of the peak(s) in Figure 14 (see Appendix A). By reducing the integral to the range 0⋯∞, the influence of poles corresponding to negative values of v has been effectively removed from the integrand v>0. Drawing on these observations, it is suggestive that the functions T1,2,3,4s,v can be separated into two components—pole terms associated with the first two terms of each, plus background terms associated with all four terms, in each of the above; it is further suggestive that only the pole terms will contribute to the integrals (see Figures 16 and 18), thereby allowing an approximate solution to be obtained. How can this be done?
Consider (66). From (68) and (69), it would appear superficially that both T1s,v and T2s,v vanish as 1/t for large values of t, which immediately demonstrates a potential inconsistency—the left-hand side of (66) vanishes exponentially and so must the right-hand side. Since the remaining integrand factors ϒR2iv and ϒI2iv contain no t dependence, the only possibility is that 1/t dependence must vanish from the (highly oscillatory) integral, and somewhere buried in the higher order asymptotic terms will be found a term with exponentially decreasing t dependence. Furthermore, the possibility also exists that cancellations will occur between the two integrand terms containing T1s,v and T2s,v in (66). Any attempt at a numerical evaluation of (66) for reasonably large values of t will immediately demonstrate the truth of this prediction in the form of a severe cancellation of significant digits. An example follows.
Chooseσ=1/3 and t=50, chosen sufficiently large to distinguish inverse powers of t from exponentially decreasing terms of the form exp−bt, but not so large that multidigit computer arithmetic must be used. For this case, calculated with 15 digits, we find(72)1π∫0∞vT113+50i,vϒR2ivdv=0.000810442386190651,(73)1π∫0∞vT213+50i,vϒI2ivdv=−0.0008104423861872670,and the sum 3.3840×10−15 is comparable with ξ1/3+50i=3.38361×10−15. Clearly, each of the integrals has an absolute value much greater than the sum of the two, and it is only because of a cancellation of digits that the final result can be found. Here, we see that the cancellation of digits has resulted in the loss of 10 digits from the sum; for larger values of t, the effect will be more significant, and one might despair of utilizing (66) for anything. It will now be shown that these (and other) cancellations can be analysed analytically, and an exponentially decreasing term can be extracted, which is the basis for the approximate model solution.
First of all, forv≤t/2, consider the asymptotic expansion t⟶∞ of each of the individual terms composing T1 labelled, respectively, by a second subscript:(74)T1,1s,v⟶−1t+2σv−4vt2+42σ−3v2+σ−12t3+Ot−4,(75)T1,2s,v⟶−1t+21−σv−4vt2+−42σ+1v2+σ2t3+Ot−4,(76)T1,3s,v⟶1t+21−σv−4vt2+42σ+1v2−σ2t3+Ot−4,(77)T1,4s,v⟶1t+2σv−4vt2+−42σ−3v2−σ−12t3+Ot−4.
Clearly, the leading asymptotic dependence1/t cancels between all of “pole” and “background” terms in pairs, demonstrating that the pole terms also contribute significantly to the asymptotic background. This verifies that a numerical evaluation of the T1s,v integral will be challenging as t increases. More importantly, it is apparent that the overall asymptotic dependence of the T1s,v part of the integral will have the leading dependence 1/t2 because the multiplicative factor ϒR2iv lacks t dependence as noted. But, as also noted, this is inconsistent with (66), where the left-hand side has an exponentially decreasing asymptotic behaviour in t. One explanation for this inconsistency could be that there exists a cancellation between the two terms in the integrand labelled by T1 and T2 to at least order 1/t2. This would explain the cancellation observed in the numerical results (72) and (73). The following gives the asymptotic expansion of each of the corresponding elements of T2s,v in analogy to (74)–(77):(78)T2,1s,v⟶−1−2vt+−4v2+σt2+−8v3+2σσ+2vt3+Ot−4,(79)T2,2s,v⟶−1−2vt+−4v2−σ+1t2+−8v3+2σ2−8σ+6vt3+Ot−4,(80)T2,3s,v⟶−1+2vt+−4v2+σt2+8v3−2σσ+2vt3+Ot−4,(81)T2,4s,v⟶−1+2vt+−4v2−σ+1t2+8v3+−2σ2+8σ−6vt3+Ot−4.
Again, we see a cancellation between each of the four terms to order1/t as well as a cancellation of the terms of order t0 with the fifth term in (69). And, again we find an inconsistency with (66) because the leading term of T2s,v is also of order 1/t2. Define each term corresponding to t−k in the asymptotic series of T1s,v by(82)T1˜k≡coefficient  oft−2kin the serieslimt⟶∞v<tT1s,v,and similarly for T2s,v. When the first two asymptotic terms of the factors in the integrand on the right-hand side of (66) are written in full, we find(83)T1s,v∼−12vt2+4v9σ2−20v2−9σ+4t4+Ot−6,(84)T2s,v∼−16v2+2t2+−64v4+48σ2−48σ+48v2−6σ2+6σ−2t4+Ot−6,so that(85)T˜11=−12vt2,(86)T˜12=4v9σ2−20v2−9σ+4t4,(87)T˜21=−16v2+2t2,(88)T˜22=−64v4+48σ2−48σ+48v2−6σ2+6σ−2t4,and clearly there can be no cancellation of terms of equal order in 1/t2, or 1/t4 between terms containing T1s,v and T2s,v. The only remaining possibility is that the integral containing each of these asymptotic forms itself vanishes identically, at least to the orders discussed above. In fact, this must be true to all asymptotic orders in t−2k,k≥0 because it is impossible for a series of the form ∑k=1∞akt−2k to add up to an exponential of the form exp−bt. Furthermore, if the integrals vanish as predicted, they must vanish for all σ. Otherwise, (66), which is an exact result cannot be asymptotically and universally true. And somewhere, accompanying all these dominant terms that vanish asymptotically to inverse polynomial order, exist other terms associated with the pole terms, that carry the exponentially vanishing result corresponding to the left hand-side of (66) in the asymptotic limit t⟶∞. Numerically, these lesser terms equate to the sum of (72) and (73). It will now be proven that the integrals do in fact vanish, as predicted, to at least the first few orders in t−2k,k=1,2,3, following which a model will be constructed to extract the terms that instead vanish exponentially.
Based on the first terms of (83) and (84), it is predicted that the sum of the integrals involving t−2 must vanish, that is,(89)1t2∫0∞−12v2ϒR2iv+v−16v2+2ϒI2ivdv=0.Proof.
Consider the classical result ([7], equation (2.15.6)) relating the inverse Mellin transform of ϒc+iv to the Jacobi Θ3 function:(90)∫−∞∞ϒc+ivdv=4π∑n=1∞e−n2x=2πΘ30,exp−πx−1,valid for c>1. We are interested in the case c=0, so shift the contour by subtracting the residue at v=1−c/i and half the residue at v=ic resulting in the identity(91)π2∑n=1∞e−n2x−πx+12=∫0∞πivx−iv+π−ivxivϒR2ivdv+i∫0∞πivx−iv−π−ivxivϒI2ivdv,if c=0, after converting the integration limits to 0,∞ and writing ϒ2iv as the sum of its real and imaginary parts. By an increasing sequence of higher order derivatives, evaluated at x=π, it is possible to obtain relevant sums using identities given by Romik [13]. The first seven such sums are listed in Appendix D, from which it is possible to obtain the first seven even moments of ϒR2iv and odd moments of ϒI2iv, listed also in Appendix D. Substituting the moments (D.2)–(D.4) into (89) verifies that the prediction (89) is true. QED.
Employing similar logic, a further prediction arises regarding the next term of order1/t4. It too must vanish for all σ. From (83), (84) and (89) we predict(92)1t4∫0∞ϒR2ivv4+v96v4−40v2−1ϒI2iv120dv=0.
Note thatσ dependence has disappeared from (92), as expected. By substituting (D.2)–(D.6) into (92) the truth of this prediction can be verified. This sequence of forecasts can be continued to the next recursive level, producing the forecast identity:(93)1t6∫0∞224ϒR2ivv6−−128v7+112v5+28v33+v3ϒI2ivdv=0.
Again,σ dependence has cancelled of its own volition. The truth of the prediction (93) can similarly be verified by substituting the set of moments (D.2)–(D.8) into (93).
To summarize, we have the following proposition.Proposition 1 (deduced and partially demonstrated).
The above sequence of predictions can be proven in general for all terms on the right-hand side of (66) that asymptotically vanish to inverse polynomial order t−2k,k>3 independently of σ.Whatever remains must then be of exponentially decreasing order and must therefore equate to the left-hand side of (66), and the challenge is to somehow isolate such term(s). The cancellation of integrals of asymptotically vanishing inverse polynomial order explains the loss of significant digits observed between (72) and (73) and suggests that such loss of significant digits will only increase for larger values of t, as the interested reader can verify for herself.Expressed another way, the crucial point is that although the peaks shown in Figure14 originate from the pole terms, these pole terms also contribute to the background terms of inverse polynomial order t−2k,k>1 that cancel. Nonetheless, the signal associated with the pole terms naturally separates from the background. The separation of signal from background is the essence of the approximate solution to be presented here.To gain further insight into these issues, consider Figure2(a) which shows (the absolute value of) the integrand:(94)J1s=σ+it,v≡T1s,vϒR2iv,as a function of v for several values of t at σ=1/3. Similarly, Figure 2(b) shows the same result for the second term in the integrand defined by(95)J2s=σ+it,v≡T2s,vϒI2iv.Three properties are immediately evident: most of the absolute value of both integrands occurs nearv=0, both J1s,v and J2s,v decrease exponentially with increasing v and are of similar magnitude—a necessity if they are to cancel numerically (see (72) and (73)). A fourth significant property is that the pole term, embedded in T1s,v at v=t/2 is clearly visible above the exponentially decreasing background in each curve of Figure 2(a), although it lies many orders of magnitude below the numerically significant portion of the integrand. In fact, it appears to be enhanced as t increases, as might be expected because the magnitude of its peak varies as t—see (C.1)—whereas the background varies as t−2k,k>1. However, in Figure 2(b) the zero of T2s,v is not readily apparent because of the resolution of each curve; moreover, each of the curves in Figure 2(b) is consistent with what appears to be the background as discussed previously. It will now be shown that these observations can be used to extract a signal from the background, and in so doing obtain an approximate solution to (66).Figure 2
The individual terms composing the integrand of (66) as a function of the integration variable v, using σ=1/3 for several values of t. Each curve has been scaled by a (smooth) factor expπt/4/t2 to separate them for easier viewing; otherwise, all of the curves coincide to a good degree of approximation, except near v≈0. (a) An illustration of the integrand J11/3+it,v at several values of t=12.5,25,50,100,200; the arrows mark the points v = t/2 that being the location of the peak of the pole term contained in T1s,v for each of the curves at each individual value of t. (b) An illustration of the integrand J21/3+it,v at several values of t=12.5,25,50,100,200; the arrows mark the points v = t/2 that being the location of the zero of the pole term contained in T2s,v for each of the curves at each individual value of t.
(a)
(b)


## 6.3. The Signal

Consider Figure3(a) which illustrates the peak of the function T1s,v over a wide range of v using s=1/3+104i. Also shown are the cumulative background terms ∑k=1NT˜1k. It can be seen that as N increases, the sum of the background terms give a better and better approximation to the background part of the function T1s,v when v<t/2. In fact, when the each of the cumulative background terms are subtracted from T1s,v, that part of the function associated with small values of v decreases (Figure 3(b)) as the number of background terms increases. But it is precisely small values of v that contribute most to the integral itself (see Figure 2). And as we have seen from (89), (92) and (93), when coupled with the full part of the integrand associated with T2s,v the integral associated with each of these background terms has either been proven, or predicted, to vanish. What is left? Only the pole term remains, wherein resides the sought after “signal.”Figure 3
The functionT11/3+104i,v in comparison with the first few cumulative background terms ∑k=1NT˜1k for wide range of v near t/2. In both figures, the vertical arrow marks the location of the peak at v=t/2. (a) An illustration of the function T11/3+104i,v as well as the cumulative first few background terms ∑k=1NT˜1k for the first 5 values of N for a wide range of v about t/2. (b) An illustration of the function T11/3+104i,v as well as the effect of subtracting the first few cumulative background terms ∑k=1NT˜1k for the first 5 values of N for a wide range of v about t/2.
(a)
(b)Remark.
Because absolute values have been used, the corresponding Figures for the functionT2s,v are qualitatively identical to Figure 3.
To understand how all this affects the integral itself, consider Figure4 which shows both integrands J1s,v and J2s,v, again using s=1/3+it with t=104, in close proximity to the peak region v=t/2. Since a logarithmic scale is used, in both figures, the alternating function(s) J12s,v are shown (dotted) colour coded, positive (red), and negative (blue). The green (dash-dot) line shows the leading asymptotic behaviour of J12s,v originating from the factor Γiv embedded in the factor ϒ2iv, scaled to fit the Figure. This is the background.
Focusing on Figure4(a), the peak of the factor T1s,v is indicated by a magenta (dashed) line, also marked by the vertical arrow. The effect of the peak of T1s,v on the integrand J1s,v can be seen as an enhancement of the integrand in the immediate vicinity of the peak. This is indicated by the superimposed solid curve representing J1s,v, again colour coded red and blue, but only for those values of t/2−2<v<t/2+2. The top horizontal dashed line equates to the ordinate of the point J1s,t/2 and the lower line delineates points lying one order of magnitude smaller that the ordinates at the peak. Figure 4(b) is similar to Figure 4(a), with the exception that instead of a peak at v=t/2, the curve representing J2s,v vanishes (see Appendix C).Figure 4
J
1
s
,
v and J2s,v in close proximity to the peak of T1s,v or zero of T2s,v, respectively, at v=t/2, as well as the shape of the background of both. (a) An illustration of the function J11/3+104i,v accompanied by T11/3+104i,v and the scaled asymptotic shape of ℜϒ11/3+104i,v very close to v = t/2. (b) An illustration of the function J21/3+104i,v accompanied by T21/3+104i,v and the scaled asymptotic shape of ℜϒ11/3+104i,v very close to v = t/2.
(a)
(b)


## 6.4. The Approximate Solution

Figures4(a) and 4(b) hold the key to the approximate solution of (66). Up to this point, all analysis has consisted of observation, (generalized) deduction, and a small amount of proof. The crucial point to be extracted from Figure 4(a) is that the peak of T1s,v “extrudes” the integrand of J1s,v above background in a small region about v=t/2. Although the background (dotted part of the curve) represents the vast majority of the magnitude of the integrand over many orders of magnitude, when this background is decomposed into terms that are of inverse polynomial order (asymptotically in t), the integration of these terms cancels between the two terms labelled by T1s,v and T2s,v and illustrated in Figures 4(a) and 4(b) (see Section 6.2), and only that part of the curve near v=t/2 (shown solid) contributes to the integral. Conversely, with reference to Figure 4(b), the strong zero of the function T2s,v acts to depress the function J2s,v near v=t/2, although the background terms still assume their role of acting to cancel similar background terms shown in Figure 4(a). All these suggest an approximation that should give a reasonable representation of a solution to (66).


### 6.4.1. Approximation 1

Let(96)Tjs,v=T˜js,v+Tojs,v,where T˜js,v represents the background, Tojs,v represents the nonbackground associated with the pole and j=1,2. From (82), we have(97)T˜js,v=∑k=1∞T˜jk.When applied to (66), this leads to(98)Js,v≡J1s,v+J2s,v=T˜1s,v+To1s,vϒRs,v+T˜2s,v+To2s,vϒIs,v=T˜1s,vϒRs,v+T˜2s,vϒIs,v+To1s,vϒRs,v+To2s,vϒIs,v.But, according to Section6.2 (i.e., Proposition 1),(99)∫0∞T˜1s,vϒRs,v+T˜2s,vϒIs,vvdv=0;thus, (66) becomes(100)ξRs=1π∫0∞T1os,vϒRs,vvdv+1π∫0∞T2os,vϒIs,vvdv.SinceTjos,v represents the nonbackground (pole) terms this leads to Approximation 2.


### 6.4.2. Approximation 2

Let(101)ξRs=1π∫t/2−δ1σ,tt/2+δ1σ,tT1os,vϒRs,vvdv+1π∫t/2−δ2σ,tt/2+δ2σ,tT2os,vϒIs,vvdv,where δjσ,t is a (small) function of Oσ0,Ot0 associated with each of Tojs,v—see (C.4); that is, only that part of the integrand in close proximity to v=t/2 contributes to the overall value of the integral, which integral is approximately symmetric about the point v=t/2. Everything else cancels. Equivalently, this recognizes that only that section of the curve coloured solid in Figures 4(a) and 4(b) will contribute to the integral, notwithstanding the fact that the dotted sections ostensibly contribute to the integrand by many more orders of magnitude. This is demonstrated numerically below (see (105)).Remark.
For clarity, I writeδjσ,t≡δj, recognizing that δjσ,t is a simple and low-order function of both σ and t; the functional dependence is investigated later.


### 6.4.3. Approximation 3

Since each of theδj are much less than any value of t in any range of interest, first rewrite (101) as(102)ξRs=δ12π∫−11T1os,t2+δ1yϒRi2δ1y+t2δ1y+tdy+δ22π∫−11T2os,t2+δ2yϒIi2δ2y+t2δ2y+tdy.By applying an obvious change of variables, expand (102) about δj=0, and consider only those terms of leading asymptotic order in t, giving the following to first order in δj and asymptotic t:(103)ξRs≈t2δ1ϒRitσσ−1π+tδ2σ2−σ−1ϒIitσσ−1π.Notice that the second term in (103) is smaller than the first term by one power of t, reflecting the fact that To2s,v vanishes at v=t/2. This means that asymptotically, we expect the second term in (103) to vanish, relative to the first term, except if ϒRit≈0.Remark.
By expanding aboutδj=0, we are selecting for values of the integrand at v=t/2. This could be interpreted as the introduction of a parameter w1 to model the width of T1os,v near its peak—effectively replacing T1os,v by inserting a box of equal area around the peak and thereby approximating parts of the integrand near v=t/2 which dominate T1os,v and thereby, the integral itself. According to the Mean Value Theorem for integrals, this can always be done and w1≠0. The parameter w1 is generalized by δ1 which captures the area of the product term T1os,vϒRiv and reasonable models for this parameter should be considered in future studies, recognizing that it is a (weak) function of both σ and t (but see Section 8). Similarly, since T2os,v vanishes antisymmetrically about v=t/2, it is reasonable to expect that it will integrate somewhat close to zero. Thus, we expect that terms involving δ2 will be much smaller than those involving δ1 which appears to be the case for large values of t. Here, let δ1=σ1−σ/2 based on numerical experimentation with the above approximation, and later, δ4=δ1. Since the relative width of the peak of T1os,v decreases for large values of t, this approximation should improve as t increases (see Figure 16).
With reference to the example presented in (72) and (73), the two terms on the right-hand side of (103) with δ1=δ2=0.2357 (=1/32) give(104)ξR13+50i≈3.63105×10−15−1.01201×10−16=3.52985×10−15,compared with the correct result ξR1/3+50i=3.3836×10−15, a reasonable approximation. Furthermore, setting δ11/3,50=δ21/3,50=0.2357 in (101) gives(105)1π∫24.7642925.2357T1os,vϒRs,vvdv+1π∫24.7642925.2357T2os,vϒIs,vvdv=2.3675×10−15+5.0984×10−16=2.8773×10−15.where I have used T1,2os,v=T1,2s,v since the latter is dominated by the pole term over the narrow range of integration (see Figure 18(a)). Note that the second term associated with δ2 is numerically smaller than the term associated with δ1 as predicted. Significantly, the order of magnitude of (105) is correct, demonstrating that the signal has been extracted, although the numerical estimate is not as good as the approximate one because the functional dependence of δ1,2 has not been generally selected for this purpose. In fact, resetting the limits of (105) using δ1=δ2=0.2983 yields the correct result to four significant figures. Of course, there is no reason to expect that δ2=δ1; this becomes less of an issue as t increases since the relative contribution from the T2os,v term asymptotically vanishes except at a zero.


### 6.4.4. Approximation 4

To obtain an asymptotic approximation ofζRσ+it as t⟶∞, first write the left-hand side of (103) in polar form and employ (7), to obtain(106)ξRs⟶−e−πt/4π−σ/2ζσ+it2πt/2σ/2−1/2σ−12tsinϕ+t2cosϕ2.Now, consider the right-hand side of (103) by identifying ϒRit in terms of its components (see (3)), apply the same approximations employed above, reunite both sides of the equation, cancelling common nonzero factors on both sides and eventually arrive at(107)ζsσ−12sinϕ+t2cosϕ≈tcosΦδ1+σ2−σ−1sinΦδ2/tπσ1−σζitt2π−σ/2.Starting from (61), a similar set of approximations can be applied. As in the model for ξRs and with reference to Figure 18, the background terms obtained from the decomposition of functions T3s,v and T4s,v integrate to zero exactly as in Section 6.2 with the change that each term contains an additional overall multiplicative factor of 2σ−1/t. Again, the dominant contribution to ξIs arises from the pole term associated with T4s,v. Thus, the equivalent of (103) is(108)ξIs≈22σ−1t2δ4ϒIit+δ3/tϒRitπσ1−σ,and the equivalent of (106) is(109)ξIs⟶e−πt/4π−σ/2ζσ+it2πt2σ/2−1/2σ−12tcosϕ−t22sinϕ.Again, because0<σ<1≪t, follow the exact steps used previously and ultimately arrive at the equivalent of (107)(110)ζs12−σcosϕ+tsinϕ≈4t1/2−σsinΦδ4+cosΦδ3/tπσ1−σζitt2π−σ/2.Remark.
Ifσ=1/2 in (110), sinϕ on the left-hand side vanishes (see (9) and Section 6.5).
Furthermore, it is possible to extract approximations forζRσ+it and ζIσ+it by solving (107) and (110) simultaneously, using a simple trigonometric expansion because argζσ+it is embedded within the definition of ϕ (see (13)), by noting that cosargζsζs=ζRs (and similarly for ζIs). Asymptotically (because both cosΦ and sinΦ cannot both vanish simultaneously), this solution yields a simple representation after applying the next approximation.


### 6.4.5. Approximation 5.

Let(111)δ2t=δ3t=0,giving(112)ζRσ+it≈2ζit−cosωcosΦδ1−2σ−1/2sinΦsinωδ4σσ−1πt2π−σ/2,(113)ζIσ+it≈2ζit−sinωcosΦδ1+2σ−1/2cosωsinΦδ4σσ−1πt2π−σ/2,where(114)ω=argΓσ2+it2−t2logπ.Notice that the right-hand sides of both (112) and (113) do not contain any components of ζσ+it, and therefore the left-hand sides are expressed in terms of independent functions (compare with [17]).


### 6.4.6. Numerical Examples

Figure5 illustrates both approximations (112) and (113) for an arbitrary choice of σ. From (103), it is seen that any structure in ξRσ+it must originate from the function ϒit on the right-hand side, and this function lacks σ dependence. Thus, (103) predicts that the structure of ξRσ+it, particularly the location of the zeros, must be relatively independent of σ since, as was noted in Section 6.1, σ dependence only resides in the transfer functions and to a lesser extent any model developed to approximate δ1,2 as a function of σ and/or t. Thus, in (66), if ϒRiv happens to vanish at v=t/2, then the peak associated with T1t,v is multiplied by a vanishing factor and contributions to the integral from both functions T1t,v=t/2 and T2t,v=t/2 vanish, independently of σ. So, the location of the zeros is approximately governed by cosΦ=0 (see the discussion surrounding (124) in Section 7). This prediction can be checked (see Figure 6(a)) which demonstrates that this property appears to be approximately true, for two randomly chosen, disparate values of σ.Figure 5
The approximations (112) and (113) to ζRσ+it and ζIσ+it, respectively, near t=500 using δ1=δ4=σ1−σ/2. (a) A comparison of the left- and right-hand sides of (112) for σ = 0.7 near t = 500. (b) A comparison of the left- and right-hand sides of (113) for σ = 0.7 near t = 500.
(a)
(b)Figure 6
For several values ofσ and t, this is a comparison of ξRσ1+it and ξRσ2+it as they exist (left) and as they are predicted to exist by the approximation (right) (see (118)). In both cases, the functions have been scaled by a factor expπt/4/t2. (a) A comparison of ξRσ+it for two very different values of σ near t = 700. (b) A comparison of the left- and right-hand sides of (118) for two very different values of σ near t = 500.
(a)
(b)Figure 7
A simple model for bothδ1σ,t and δ4σ,t exists as a function of σ, although the dependence on t may not be simple (see Figure 10(a)). (a) The solution function Δ1σ,t for several values of t over a range of σ. Each of the curves has been normalized by Δ11/2,t with values shown. (b) The solution function Δ4σ,t for several values of t over a range of σ. Each of the curves has been normalized by Δ41/2,t with values shown.
(a)
(b)From this observation, arises a constraint on the approximation. Consider two instances of (103) with different choices σ=σ1 and σ=σ2. The approximation predicts that(115)ξRσ1+itξRσ2+it=δ1σ1σ21−σ2δ1σ2σ11−σ1,since the model suggests that δ1 is at least a (weak) function of σ. Now, suppose that σ2=1−σ1, in which case the left-hand side of (115) becomes unity (see (2)); for consistency, this imposes a constraint on the parameter δ1, specifically(116)δ1σ=δ11−σ.With this constraint in mind, and consistent with the approximations specified previously, it is possible to obtain some insight into a reasonable model forδ1σ,t by solving (103) using the approximation δ2/t=0 (see Approximation 5). Introduce σ,t dependence by letting the solution be denoted by Δ1σ,t and consider Δ1σ,t over the range 0≤σ≤1, for several choices of t. The result is given in Figure 7(a), providing evidence that it is likely that a smooth function exists to capture the σ dependence of δ1σ,t, although the normalization factor varies considerably as a function of t. The somewhat arbitrary and simple model(117)δ1σ,t=σ1−σ2,used throughout this paper is also shown; it is clear that this model, although a reasonable first approximation, requires future refinement. A similar result can be found for Δ4σ,t, the solution of (108) (see Figure 7(b)). (Remark: to obtain the normalization of the latter figure, expand the function ξσ+it about the point σ=1/2 and cancel the numerator and denominator factors σ−1/2 that appear in the imaginary part).Figure 8
A comparison of the left- and right-hand sides of (107) for two very different values of σ near t=900. The parameter δ1=σ1−σ/2 in both cases, and all curves have been normalized by the factor 1/t.Figure 9
A comparison of the left- and right-hand sides of (119) near t=900. The parameter δ1=σ1−σ/2 and σ=1/2.All this suggests that it is of interest to investigateξσ+it for two different choices of σ. (115) and (117) predict that(118)σ11−σ1ξRσ1+it≈ξRσ2+itσ21−σ2,an example of which is shown in Figure 6(b). Finally, it is of interest to consider approximations to ζσ+it as given in (107) (see Figure 8).Figure 10
A test (a) of the solution to (121) over the range 700≤t≤720 applied (b) to that same equation over a completely different range of t. (a) The curve in this Figure shows the solution Δ41/2,t of (121) over a range of t. The horizontal line Δ41/2,t≈0.16 illustrates a first approximation. (b) This is a comparison of the absolute value of the right and left-hand sides of (120) using Δ41/2,t=0.16.
(a)
(b)


## 6.4.1. Approximation 1

Let(96)Tjs,v=T˜js,v+Tojs,v,where T˜js,v represents the background, Tojs,v represents the nonbackground associated with the pole and j=1,2. From (82), we have(97)T˜js,v=∑k=1∞T˜jk.When applied to (66), this leads to(98)Js,v≡J1s,v+J2s,v=T˜1s,v+To1s,vϒRs,v+T˜2s,v+To2s,vϒIs,v=T˜1s,vϒRs,v+T˜2s,vϒIs,v+To1s,vϒRs,v+To2s,vϒIs,v.But, according to Section6.2 (i.e., Proposition 1),(99)∫0∞T˜1s,vϒRs,v+T˜2s,vϒIs,vvdv=0;thus, (66) becomes(100)ξRs=1π∫0∞T1os,vϒRs,vvdv+1π∫0∞T2os,vϒIs,vvdv.SinceTjos,v represents the nonbackground (pole) terms this leads to Approximation 2.


## 6.4.2. Approximation 2

Let(101)ξRs=1π∫t/2−δ1σ,tt/2+δ1σ,tT1os,vϒRs,vvdv+1π∫t/2−δ2σ,tt/2+δ2σ,tT2os,vϒIs,vvdv,where δjσ,t is a (small) function of Oσ0,Ot0 associated with each of Tojs,v—see (C.4); that is, only that part of the integrand in close proximity to v=t/2 contributes to the overall value of the integral, which integral is approximately symmetric about the point v=t/2. Everything else cancels. Equivalently, this recognizes that only that section of the curve coloured solid in Figures 4(a) and 4(b) will contribute to the integral, notwithstanding the fact that the dotted sections ostensibly contribute to the integrand by many more orders of magnitude. This is demonstrated numerically below (see (105)).Remark.
For clarity, I writeδjσ,t≡δj, recognizing that δjσ,t is a simple and low-order function of both σ and t; the functional dependence is investigated later.


## 6.4.3. Approximation 3

Since each of theδj are much less than any value of t in any range of interest, first rewrite (101) as(102)ξRs=δ12π∫−11T1os,t2+δ1yϒRi2δ1y+t2δ1y+tdy+δ22π∫−11T2os,t2+δ2yϒIi2δ2y+t2δ2y+tdy.By applying an obvious change of variables, expand (102) about δj=0, and consider only those terms of leading asymptotic order in t, giving the following to first order in δj and asymptotic t:(103)ξRs≈t2δ1ϒRitσσ−1π+tδ2σ2−σ−1ϒIitσσ−1π.Notice that the second term in (103) is smaller than the first term by one power of t, reflecting the fact that To2s,v vanishes at v=t/2. This means that asymptotically, we expect the second term in (103) to vanish, relative to the first term, except if ϒRit≈0.Remark.
By expanding aboutδj=0, we are selecting for values of the integrand at v=t/2. This could be interpreted as the introduction of a parameter w1 to model the width of T1os,v near its peak—effectively replacing T1os,v by inserting a box of equal area around the peak and thereby approximating parts of the integrand near v=t/2 which dominate T1os,v and thereby, the integral itself. According to the Mean Value Theorem for integrals, this can always be done and w1≠0. The parameter w1 is generalized by δ1 which captures the area of the product term T1os,vϒRiv and reasonable models for this parameter should be considered in future studies, recognizing that it is a (weak) function of both σ and t (but see Section 8). Similarly, since T2os,v vanishes antisymmetrically about v=t/2, it is reasonable to expect that it will integrate somewhat close to zero. Thus, we expect that terms involving δ2 will be much smaller than those involving δ1 which appears to be the case for large values of t. Here, let δ1=σ1−σ/2 based on numerical experimentation with the above approximation, and later, δ4=δ1. Since the relative width of the peak of T1os,v decreases for large values of t, this approximation should improve as t increases (see Figure 16).
With reference to the example presented in (72) and (73), the two terms on the right-hand side of (103) with δ1=δ2=0.2357 (=1/32) give(104)ξR13+50i≈3.63105×10−15−1.01201×10−16=3.52985×10−15,compared with the correct result ξR1/3+50i=3.3836×10−15, a reasonable approximation. Furthermore, setting δ11/3,50=δ21/3,50=0.2357 in (101) gives(105)1π∫24.7642925.2357T1os,vϒRs,vvdv+1π∫24.7642925.2357T2os,vϒIs,vvdv=2.3675×10−15+5.0984×10−16=2.8773×10−15.where I have used T1,2os,v=T1,2s,v since the latter is dominated by the pole term over the narrow range of integration (see Figure 18(a)). Note that the second term associated with δ2 is numerically smaller than the term associated with δ1 as predicted. Significantly, the order of magnitude of (105) is correct, demonstrating that the signal has been extracted, although the numerical estimate is not as good as the approximate one because the functional dependence of δ1,2 has not been generally selected for this purpose. In fact, resetting the limits of (105) using δ1=δ2=0.2983 yields the correct result to four significant figures. Of course, there is no reason to expect that δ2=δ1; this becomes less of an issue as t increases since the relative contribution from the T2os,v term asymptotically vanishes except at a zero.


## 6.4.4. Approximation 4

To obtain an asymptotic approximation ofζRσ+it as t⟶∞, first write the left-hand side of (103) in polar form and employ (7), to obtain(106)ξRs⟶−e−πt/4π−σ/2ζσ+it2πt/2σ/2−1/2σ−12tsinϕ+t2cosϕ2.Now, consider the right-hand side of (103) by identifying ϒRit in terms of its components (see (3)), apply the same approximations employed above, reunite both sides of the equation, cancelling common nonzero factors on both sides and eventually arrive at(107)ζsσ−12sinϕ+t2cosϕ≈tcosΦδ1+σ2−σ−1sinΦδ2/tπσ1−σζitt2π−σ/2.Starting from (61), a similar set of approximations can be applied. As in the model for ξRs and with reference to Figure 18, the background terms obtained from the decomposition of functions T3s,v and T4s,v integrate to zero exactly as in Section 6.2 with the change that each term contains an additional overall multiplicative factor of 2σ−1/t. Again, the dominant contribution to ξIs arises from the pole term associated with T4s,v. Thus, the equivalent of (103) is(108)ξIs≈22σ−1t2δ4ϒIit+δ3/tϒRitπσ1−σ,and the equivalent of (106) is(109)ξIs⟶e−πt/4π−σ/2ζσ+it2πt2σ/2−1/2σ−12tcosϕ−t22sinϕ.Again, because0<σ<1≪t, follow the exact steps used previously and ultimately arrive at the equivalent of (107)(110)ζs12−σcosϕ+tsinϕ≈4t1/2−σsinΦδ4+cosΦδ3/tπσ1−σζitt2π−σ/2.Remark.
Ifσ=1/2 in (110), sinϕ on the left-hand side vanishes (see (9) and Section 6.5).
Furthermore, it is possible to extract approximations forζRσ+it and ζIσ+it by solving (107) and (110) simultaneously, using a simple trigonometric expansion because argζσ+it is embedded within the definition of ϕ (see (13)), by noting that cosargζsζs=ζRs (and similarly for ζIs). Asymptotically (because both cosΦ and sinΦ cannot both vanish simultaneously), this solution yields a simple representation after applying the next approximation.


## 6.4.5. Approximation 5.

Let(111)δ2t=δ3t=0,giving(112)ζRσ+it≈2ζit−cosωcosΦδ1−2σ−1/2sinΦsinωδ4σσ−1πt2π−σ/2,(113)ζIσ+it≈2ζit−sinωcosΦδ1+2σ−1/2cosωsinΦδ4σσ−1πt2π−σ/2,where(114)ω=argΓσ2+it2−t2logπ.Notice that the right-hand sides of both (112) and (113) do not contain any components of ζσ+it, and therefore the left-hand sides are expressed in terms of independent functions (compare with [17]).


## 6.4.6. Numerical Examples

Figure5 illustrates both approximations (112) and (113) for an arbitrary choice of σ. From (103), it is seen that any structure in ξRσ+it must originate from the function ϒit on the right-hand side, and this function lacks σ dependence. Thus, (103) predicts that the structure of ξRσ+it, particularly the location of the zeros, must be relatively independent of σ since, as was noted in Section 6.1, σ dependence only resides in the transfer functions and to a lesser extent any model developed to approximate δ1,2 as a function of σ and/or t. Thus, in (66), if ϒRiv happens to vanish at v=t/2, then the peak associated with T1t,v is multiplied by a vanishing factor and contributions to the integral from both functions T1t,v=t/2 and T2t,v=t/2 vanish, independently of σ. So, the location of the zeros is approximately governed by cosΦ=0 (see the discussion surrounding (124) in Section 7). This prediction can be checked (see Figure 6(a)) which demonstrates that this property appears to be approximately true, for two randomly chosen, disparate values of σ.Figure 5
The approximations (112) and (113) to ζRσ+it and ζIσ+it, respectively, near t=500 using δ1=δ4=σ1−σ/2. (a) A comparison of the left- and right-hand sides of (112) for σ = 0.7 near t = 500. (b) A comparison of the left- and right-hand sides of (113) for σ = 0.7 near t = 500.
(a)
(b)Figure 6
For several values ofσ and t, this is a comparison of ξRσ1+it and ξRσ2+it as they exist (left) and as they are predicted to exist by the approximation (right) (see (118)). In both cases, the functions have been scaled by a factor expπt/4/t2. (a) A comparison of ξRσ+it for two very different values of σ near t = 700. (b) A comparison of the left- and right-hand sides of (118) for two very different values of σ near t = 500.
(a)
(b)Figure 7
A simple model for bothδ1σ,t and δ4σ,t exists as a function of σ, although the dependence on t may not be simple (see Figure 10(a)). (a) The solution function Δ1σ,t for several values of t over a range of σ. Each of the curves has been normalized by Δ11/2,t with values shown. (b) The solution function Δ4σ,t for several values of t over a range of σ. Each of the curves has been normalized by Δ41/2,t with values shown.
(a)
(b)From this observation, arises a constraint on the approximation. Consider two instances of (103) with different choices σ=σ1 and σ=σ2. The approximation predicts that(115)ξRσ1+itξRσ2+it=δ1σ1σ21−σ2δ1σ2σ11−σ1,since the model suggests that δ1 is at least a (weak) function of σ. Now, suppose that σ2=1−σ1, in which case the left-hand side of (115) becomes unity (see (2)); for consistency, this imposes a constraint on the parameter δ1, specifically(116)δ1σ=δ11−σ.With this constraint in mind, and consistent with the approximations specified previously, it is possible to obtain some insight into a reasonable model forδ1σ,t by solving (103) using the approximation δ2/t=0 (see Approximation 5). Introduce σ,t dependence by letting the solution be denoted by Δ1σ,t and consider Δ1σ,t over the range 0≤σ≤1, for several choices of t. The result is given in Figure 7(a), providing evidence that it is likely that a smooth function exists to capture the σ dependence of δ1σ,t, although the normalization factor varies considerably as a function of t. The somewhat arbitrary and simple model(117)δ1σ,t=σ1−σ2,used throughout this paper is also shown; it is clear that this model, although a reasonable first approximation, requires future refinement. A similar result can be found for Δ4σ,t, the solution of (108) (see Figure 7(b)). (Remark: to obtain the normalization of the latter figure, expand the function ξσ+it about the point σ=1/2 and cancel the numerator and denominator factors σ−1/2 that appear in the imaginary part).Figure 8
A comparison of the left- and right-hand sides of (107) for two very different values of σ near t=900. The parameter δ1=σ1−σ/2 in both cases, and all curves have been normalized by the factor 1/t.Figure 9
A comparison of the left- and right-hand sides of (119) near t=900. The parameter δ1=σ1−σ/2 and σ=1/2.All this suggests that it is of interest to investigateξσ+it for two different choices of σ. (115) and (117) predict that(118)σ11−σ1ξRσ1+it≈ξRσ2+itσ21−σ2,an example of which is shown in Figure 6(b). Finally, it is of interest to consider approximations to ζσ+it as given in (107) (see Figure 8).Figure 10
A test (a) of the solution to (121) over the range 700≤t≤720 applied (b) to that same equation over a completely different range of t. (a) The curve in this Figure shows the solution Δ41/2,t of (121) over a range of t. The horizontal line Δ41/2,t≈0.16 illustrates a first approximation. (b) This is a comparison of the absolute value of the right and left-hand sides of (120) using Δ41/2,t=0.16.
(a)
(b)


## 6.5. The Caseσ=1/2

From the real part ofξs, (107) is of particular interest when σ=1/2. By straight substitution, using (117) and (8), we obtain(119)ζ12+itcosϕ≈4δ112,tcosΦζ1−itt1/423/4π5/4.This is consistent with equation (5.1.8) in [7]; for a numerical example, see Figure 9. In (119), note that the factor cosϕ=−1k because of (13), where k approximates the number of zeros of ζ1/2+it relative to t=0. For further discussion, see Section 8.2.From the imaginary part ofξs, in the case of (108) with σ=1/2, both sides vanish, as they must because it is known that ξI1/2+it=0. This is also true of (110) because in the limit σ=1/2, we have ϕ=kπ due to (9) and (13). From (10), by straightforward differentiation of (2) applied to (108), we obtain(120)limσ⟶1/2ξI1/2+itσ−1/2=ℑ∂∂σξσ+itσ=1/2=−1kΓ1/4+it/2π1/4t−14t2+14ℑψ14+it2ζ12+it+12t2+14sinα−βζ′12+it≈4t2δ4ϒIit+δ3/tϒRitπσ1−σ,where ψs is the digamma function and α and β are defined in (16) and (17), respectively. Asymptotically this becomes(121)2π3/412ψI14+it2ζ12+it−sinα−βζ′12+it≈−64δ4ζitsinΦt−1/4.Figure 11
A comparison of the predicted relationship betweenξRσ+it (left) and ξIσ+it (right) versus the corresponding real and imaginary components of ϒit at two different choices of σ near t=800. The figures are scaled by eπt/4/t2. (a) A comparison of the left- and right-hand sides of (103) near t = 800. The parameter δ1 =σ1−σ/2, σ=1/2 and δ2=0. (b) A comparison of the left- and right-hand sides of (108) near t = 800. The parameter δ4 =σ1−σ/2, σ=0.22 and δ3=0.
(a)
(b)For this fixed value ofσ=1/2, it is possible to obtain an estimate of δ41/2,t by solving (121) for δ4. Denote the solution by Δ41/2,t and consider Figure 10(a) which shows Δ41/2,t for a range of t. As a first approximation, choose δ41/2,t=0.16 as suggested in the figure. It should be noted that the denominator of the solution can vanish, but the numerator does not vanish simultaneously; these singular points of the solution are apparent in the figure; however, at such points, the parameter δ31/2,t must be incorporated into the model; this will prevent the singularity from occurring. Thus, the approximate solution in Figure 10(a) is valid in between the singular points; the horizontal line Δ41/2,t≈0.16 used in this example has been chosen on that basis. Figure 10(b) uses δ4 = .16 and δ3 = 0 to compare the right- and left-hand sides of (120).


## 7. Zeros ofζσ+it

As noted earlier,c=−1 was chosen in (59) because it would lead to an expression involving a master function ζ1−it (see (8)) whose properties are fairly well known and whose validity spans the critical strip. In particular, it is known that ζ1−it≠0 (see [18], Theorem (6.1)), although both ζR1−it and ζI1−it do have many (noncoincident) zeros, and their location is the subject of some interest (e.g., [19]). Furthermore, it is well known that ξs shares the zeros of ζs and ϒ1−it shares the zeros of ζ1−it, so any properties of the zeros belonging to the first of each pair, apply to the second (of each pair).Therefore, consider (103) and (108) in the company of Approximation 5. The first of these relates ξRs to ϒRit and predicts that the zeros of these functions should coincide, and, in fact, the two should be proportional to one another if σ≠0 and σ≠1 where the model c=−1 breaks down anyway. The second of these relates ξIs to ϒIit and predicts a further coincidence between the zeros of ξIσ+it and ϒIit unless σ=1/2, in which case ξI1/2+it is always zero ([1], Section 6.5); ϒIit is therefore independent if σ=1/2. But, as noted, because ζ1−it≠0 then no zeros of ϒRit and ϒIit coincide (see also (8)). Consequently, neither do the zeros of ξRσ+it and ξIσ+it. Furthermore, if ξσ+it=0 and σ≠1/2, this would require the right-hand sides of (103) and (108) to vanish simultaneously, a contradiction with the known properties of ζ1−it because neither of the two parameters δ1 and δ4 vanish.Therefore, ifσ≠1/2, it is impossible for ξσ+it to vanish because it is overly constrained by the properties of ζ1−it; if σ=1/2, one of the constraints vanishes, and ξ1/2+it=0 becomes possible. Thus, within the confines of this approximation, the Riemann hypothesis (RH) is true. However, none of the above predicts that ζ1/2+it=0 anywhere. That is a separate issue. A comparison between the right- and left-hand sides of (103) and (108) is given in Figure 11, demonstrating that the discussion here closely approximates reality.Figure 12
A test of (124) when σ=1/2 near t=900. The vertical lines mark the location of the points t0, where ζ1/2+it0=0, theorized to coincide with cosΦt0=0.It is also worthwhile to consider how (112) and (113) allow us to reach the same conclusion. Suppose that ζRσ+it=0 on the left-hand side of (112) and simultaneously ζIσ+it=0 on the left-hand side of (113). Since it is known that ζ1−it≠0 (and therefore, ζit≠0) it must be that each of the terms in parenthesis in those two equations vanish simultaneously. That is, after removing all nonzero common factors, and noting that the equations are only valid for 0<σ<1 we have(122)2−1+2σsinωδ4sinΦ1−σ+2cosΦcosωδ121−σ=0,2−1+2σcosωδ4sinΦ1−σ−2sinωδ1cosΦ21−σ=0.Ifσ≠1/2, if cosΦ≠0, the only solution to (122) requires that(123)cos2ω+sin2ω=0,which is obviously impossible. If cosΦ=0, a second formal solution exists; it requires that either sinΦ=0 also or that sinω and cosω vanishes simultaneously. Both of these conditions are incompatible. Therefore, ζs≠0 if σ≠1/2. However, in the case that σ=1/2, the only solution to (122) is(124)cosΦt=0.which is entirely possible and consistent with everything else that has been deduced so far (see Figure 12).Notice that ift=t0 defines a zero on the critical line, the term ζ1/2+it0 on the left-hand side of (121) vanishes, and since it is known ([20], equation (4.29) and Figure 3) that at a zero defined by t=t0, the argument functions(125)αt0−βt0=±π2,then the accuracy of the approximation δ41/2,t0=0.16 can be tested, because (121) becomes(126)ζ′12+itt=t0=2.17ζitsinϕit2/t1/4t=t0.A numerical exploration of (126) is given in Figure 13 where the absolute ratio of the left- and right-hand sides of (126) is shown. The vertical lines mark each of the zeros t=t0 for the range of t under consideration, and if (126) is true, the solid (blue) curve should cross each vertical line with the value of unity, indicated by the dotted horizontal line.Figure 13
Ratio of the left and absolute value of the right-hand sides of (126) for general values of t>0. Vertical lines mark the location of the zeros.Finally, it needs to be mentioned that the above discussion did not consider any detailed functional dependence ofδ1 and δ4. Both these parameters were introduced to model the width of the peaks (i.e., extrusions) in the products T1s,vϒRiv and T4s,vϒIiv, and it has been shown in the Appendices that the underlying widths of T1s,v and T4s,v never vanish. Therefore, neither do δ1 and δ4, and so neither affects the discussion regarding the solution of (122) and the existence of the zeros.


## 8. Caveats

### 8.1. Universality

#### 8.1.1. A Correspondent Has Written

Unfortunately the fundamental flaw of the paper is the premise that (approximately) the zeta function ats=σ+it is related in an elementary way to the same function at z=σ0+it (i.e., for the same t but different σ). This contradicts Voronin’s Universality Theorem, which basically says the zeta function, sufficiently far up the critical strip, mimics (to within arbitrary accuracy) all non-vanishing analytic functions in simply connected compact subsets of the strip 1/2<σ<1. So, while the behaviour of ζσ+it may appear to be quite regular for 1/2<σ<1 and fixed t of moderate size (indeed monotonic for real, imaginary and absolute values), this cannot be true as one moves further up the strip. Indeed, one can choose an arbitrary “wild” nonvanishing analytic function, and an arbitrary ϵ>0 , such that the zeta function approximates that function to within ϵ for some (probably extremely large) fixed t and 1/2<σ<1. Therefore, approximations of the form (107) and (119) cannot hold uniformly for large t.


#### 8.1.2. The Author Comments

The fundamental result of this paper is a proof thatζs1 is related to the same function ζs2 integrated over a contour through the intermediary of a specific integral operator. This is a rigorous result, which, as noted earlier, has been shown ([15], Theorem 1) to be equivalent to the Cauchy Contour integral theorem. In this paper, a specific example has been chosen such that s2 covers the 1-line through the intermediary of the functional equation (8). Subject to Proposition 1, the representation (101) introduces arbitrary functions δ1,2,3,4σ,t, and the remainder of the paper investigates the observation that if these functions are chosen in an elementary manner, a reasonable representation of the function ζσ+it can be found with reference to the function ζit (and, by reflection, ζ1−it) for 0<σ<1. Surprisingly, the numerical examples presented suggest that the connection between the values of ζσ+it and ζit are remarkably insensitive to the particular model used for δ1,4σ,t. This was investigated in a superficial fashion (see Figures 7 and 10(a)); a detailed analysis is beyond the scope of this paper. Certainly, if at an extremely large value of t the zeta function assumes a “wild” value, the functions δ1,2,3,4σ,t could be chosen appropriately to satisfy whatever wildness arises. However, the purpose of an approximate solution to a complex problem is to convert chaos into order, not the reverse.Therefore, notice that the correspondent, perhaps inadvertently, raises an interesting point: in addition to the Universality Theorem, (see ([7], Chapter XI)), the function ξs (and hence, ζs) must also obey restraints embedded in the newly derived integral equations (61) and (66). Consider the consequence(s), taking note that the following results are all rigorously proven and widely accepted:(i)
The Universality Theorem recognizes that somewhere “further up the strip,” the functionζs will assume a “wild” value that approximates an arbitrary function within ϵ.(ii)
Equations (61) and (66) impose a strict relationship between ξs and ξ2iv on the s=0+2iv line, and therefore a relationship exists between ζs and ζ1−2iv through these equations (2) and the functional equation.(iii)
The functionζ1−2iv appearing in (61) and (66) via the functional equation and its polar-form is bounded by known limits on ζ1−2iv through well-established theorems. Crudely ζ1−2iv≪log2/32v, as well as other more strict limits given in [16] and [21] (see (131) and (132)).(iv)
The transfer functions embedded in (61) and (66) are simple polynomials whose maximum t−value anywhere on the contour is limited to Ot1, and these functions are the only elements in the integrand(s) that carry t dependence.(v)
The only other significant function appearing in the integrand of (61) and (66) is the product Γivπ−iv whose smoothness (continuity) and magnitude properties are well-known.(vi)
None of the elements of the integrand in (61) and (66) contain real poles or divergences and the integrals themselves are obviously convergent at both endpoints.(vii)
Consequently, the integrals in (61) and (66) are all limited in magnitude and constrained in their t−dependence and so cannot assume an arbitrarily “wild” value or approximate an arbitrarily “wild” function. This is a rigorous result contingent on the latter four results cited above, all of which are rigorous and true both in their own right as well as more general instances corresponding to any c>−1 (see Section 4).All of this suggests that an incompatibility exists among one or more of the above-cited rigorous results. Specifically, it is not apparent how an arbitrarily “wild” function, whose magnitude exceeds the known maximum value of the product of all the components of the integrands of (61) and (66) integrated over the range 0,∞, could be approximated “within ϵ” by ζσ+it, which is subject to the integral equation constraint(s) specified in the above. Furthermore, the correspondent’s observation that the approximate solutions (107) and (119) do not satisfy the Universality Theorem suggests that their precursor—the (rigorously proven) integral equation itself—does not satisfy the Universality Theorem. In summary, the reconciliation of the noted apparent incompatibility suggests a number of fruitful subjects for future investigation—again, beyond the scope of this work.


### 8.2. Lindelöf’s Hypothesis

#### 8.2.1. A Second Correspondent Has Written

“(1) seems to indicate that t−1/4ζ1/2+it is unbounded as t⟶∞ which we know is not true; see [22]. In fact, the Lindelöf hypothesis states that ζ1/2+it=Otϵ as t⟶∞ for all ϵ>0, and if this is false, so is the Riemann hypothesis.”


#### 8.2.2. The Author Comments

According to Edwards ([1], Section 9.2), Lindelöf’s theorem is based on Lindelöf’s observation that “there is a constant K such that(127)ζσ+it<Kt1/2−σ/2logt,throughout the half-strip 0≤ℜs≤1.” Assuming that ζ1/2+it asymptotically obeys a power law, Lindelöf denotes by μσ the least upper bound of the numbers A such that ζσ+itt−A is bounded as t⟶∞, shows that(128)0≤μ12≤14,and hypothesizes that μ1/2=0. This basically posits that as t⟶∞, ζ1/2+it increases at least faster than Ot0. Several proven refinements of (128) are known, over the years gradually reducing the upper inequality limit from 1/4 to the most recent [22] value 13/84=0.1548, yielding the least upper asymptotic bound(129)ζ12+it≪Ot13/84.By comparison, the equivalent result obtained here (see (119)) is(130)ζ12+itcosϕ≈4δ112,tcosΦtζ1−itt1/423/4π5/4,where cosϕ=−1k counts the zeros and −1≤cosΦt≤1. For emphasis, Φ has been written fully as Φt (see (15)). With ([21], Conjecture A)(131)maxt∈1,Tζ1+it∼expγlog logT,or ([16], Theorem 1)(132)maxt∈T,Tζ1+it≥expγlog logT+log log logT−C,both of which are valid for sufficiently large T, (130) becomes(133)ζ12+itcosϕ≈4δ112,tcosΦtt1/423/4π5/4loglogt+⋯.Notice the similarity between (107) (with (8)) and (127) for general σ, as well as (127) and (133) when σ=1/2. Up to this point, the functions δ1,2,3,4σ,t have been simply modelled as a function of σ, although clearly t dependence exists, e.g., see Figure 10(a). So, considering that δ11/2,t is expected to be a weak function of t as noted previously, it is reasonable to anticipate that t dependence residing in δ1σ,t and to a lesser extent cosΦt could affect any (dis)agreement between (127), (129), (133), and Lindelöf’s hypothesis.In contrast to Section8.1, where an arbitrary “wildness” was needed to generate agreement, here it is noted that the functions δ1,21/2,t capture the value of the integrand of the integral equation only in close proximity to the point v=t/2 in (105). Thus t-dependence of δ1,3 and δ2,4 is sensitive to the density of the real and imaginary zeros (respectively) of the function ζ1−2iv near a particular point on the 1-line. And, as previously noted, such t−dependence must persist for any c>−1, in which case it is useful to recall [19] that ℜζσ+it>0 if σ>1.192... corresponding to c>−0.904.... No arbitrariness here. In any case, it is evident that investigations into these observations might bear fruit. Again, this is beyond the scope of this work.


## 8.1. Universality

### 8.1.1. A Correspondent Has Written

Unfortunately the fundamental flaw of the paper is the premise that (approximately) the zeta function ats=σ+it is related in an elementary way to the same function at z=σ0+it (i.e., for the same t but different σ). This contradicts Voronin’s Universality Theorem, which basically says the zeta function, sufficiently far up the critical strip, mimics (to within arbitrary accuracy) all non-vanishing analytic functions in simply connected compact subsets of the strip 1/2<σ<1. So, while the behaviour of ζσ+it may appear to be quite regular for 1/2<σ<1 and fixed t of moderate size (indeed monotonic for real, imaginary and absolute values), this cannot be true as one moves further up the strip. Indeed, one can choose an arbitrary “wild” nonvanishing analytic function, and an arbitrary ϵ>0 , such that the zeta function approximates that function to within ϵ for some (probably extremely large) fixed t and 1/2<σ<1. Therefore, approximations of the form (107) and (119) cannot hold uniformly for large t.


### 8.1.2. The Author Comments

The fundamental result of this paper is a proof thatζs1 is related to the same function ζs2 integrated over a contour through the intermediary of a specific integral operator. This is a rigorous result, which, as noted earlier, has been shown ([15], Theorem 1) to be equivalent to the Cauchy Contour integral theorem. In this paper, a specific example has been chosen such that s2 covers the 1-line through the intermediary of the functional equation (8). Subject to Proposition 1, the representation (101) introduces arbitrary functions δ1,2,3,4σ,t, and the remainder of the paper investigates the observation that if these functions are chosen in an elementary manner, a reasonable representation of the function ζσ+it can be found with reference to the function ζit (and, by reflection, ζ1−it) for 0<σ<1. Surprisingly, the numerical examples presented suggest that the connection between the values of ζσ+it and ζit are remarkably insensitive to the particular model used for δ1,4σ,t. This was investigated in a superficial fashion (see Figures 7 and 10(a)); a detailed analysis is beyond the scope of this paper. Certainly, if at an extremely large value of t the zeta function assumes a “wild” value, the functions δ1,2,3,4σ,t could be chosen appropriately to satisfy whatever wildness arises. However, the purpose of an approximate solution to a complex problem is to convert chaos into order, not the reverse.Therefore, notice that the correspondent, perhaps inadvertently, raises an interesting point: in addition to the Universality Theorem, (see ([7], Chapter XI)), the function ξs (and hence, ζs) must also obey restraints embedded in the newly derived integral equations (61) and (66). Consider the consequence(s), taking note that the following results are all rigorously proven and widely accepted:(i)
The Universality Theorem recognizes that somewhere “further up the strip,” the functionζs will assume a “wild” value that approximates an arbitrary function within ϵ.(ii)
Equations (61) and (66) impose a strict relationship between ξs and ξ2iv on the s=0+2iv line, and therefore a relationship exists between ζs and ζ1−2iv through these equations (2) and the functional equation.(iii)
The functionζ1−2iv appearing in (61) and (66) via the functional equation and its polar-form is bounded by known limits on ζ1−2iv through well-established theorems. Crudely ζ1−2iv≪log2/32v, as well as other more strict limits given in [16] and [21] (see (131) and (132)).(iv)
The transfer functions embedded in (61) and (66) are simple polynomials whose maximum t−value anywhere on the contour is limited to Ot1, and these functions are the only elements in the integrand(s) that carry t dependence.(v)
The only other significant function appearing in the integrand of (61) and (66) is the product Γivπ−iv whose smoothness (continuity) and magnitude properties are well-known.(vi)
None of the elements of the integrand in (61) and (66) contain real poles or divergences and the integrals themselves are obviously convergent at both endpoints.(vii)
Consequently, the integrals in (61) and (66) are all limited in magnitude and constrained in their t−dependence and so cannot assume an arbitrarily “wild” value or approximate an arbitrarily “wild” function. This is a rigorous result contingent on the latter four results cited above, all of which are rigorous and true both in their own right as well as more general instances corresponding to any c>−1 (see Section 4).All of this suggests that an incompatibility exists among one or more of the above-cited rigorous results. Specifically, it is not apparent how an arbitrarily “wild” function, whose magnitude exceeds the known maximum value of the product of all the components of the integrands of (61) and (66) integrated over the range 0,∞, could be approximated “within ϵ” by ζσ+it, which is subject to the integral equation constraint(s) specified in the above. Furthermore, the correspondent’s observation that the approximate solutions (107) and (119) do not satisfy the Universality Theorem suggests that their precursor—the (rigorously proven) integral equation itself—does not satisfy the Universality Theorem. In summary, the reconciliation of the noted apparent incompatibility suggests a number of fruitful subjects for future investigation—again, beyond the scope of this work.


## 8.1.1. A Correspondent Has Written

Unfortunately the fundamental flaw of the paper is the premise that (approximately) the zeta function ats=σ+it is related in an elementary way to the same function at z=σ0+it (i.e., for the same t but different σ). This contradicts Voronin’s Universality Theorem, which basically says the zeta function, sufficiently far up the critical strip, mimics (to within arbitrary accuracy) all non-vanishing analytic functions in simply connected compact subsets of the strip 1/2<σ<1. So, while the behaviour of ζσ+it may appear to be quite regular for 1/2<σ<1 and fixed t of moderate size (indeed monotonic for real, imaginary and absolute values), this cannot be true as one moves further up the strip. Indeed, one can choose an arbitrary “wild” nonvanishing analytic function, and an arbitrary ϵ>0 , such that the zeta function approximates that function to within ϵ for some (probably extremely large) fixed t and 1/2<σ<1. Therefore, approximations of the form (107) and (119) cannot hold uniformly for large t.


## 8.1.2. The Author Comments

The fundamental result of this paper is a proof thatζs1 is related to the same function ζs2 integrated over a contour through the intermediary of a specific integral operator. This is a rigorous result, which, as noted earlier, has been shown ([15], Theorem 1) to be equivalent to the Cauchy Contour integral theorem. In this paper, a specific example has been chosen such that s2 covers the 1-line through the intermediary of the functional equation (8). Subject to Proposition 1, the representation (101) introduces arbitrary functions δ1,2,3,4σ,t, and the remainder of the paper investigates the observation that if these functions are chosen in an elementary manner, a reasonable representation of the function ζσ+it can be found with reference to the function ζit (and, by reflection, ζ1−it) for 0<σ<1. Surprisingly, the numerical examples presented suggest that the connection between the values of ζσ+it and ζit are remarkably insensitive to the particular model used for δ1,4σ,t. This was investigated in a superficial fashion (see Figures 7 and 10(a)); a detailed analysis is beyond the scope of this paper. Certainly, if at an extremely large value of t the zeta function assumes a “wild” value, the functions δ1,2,3,4σ,t could be chosen appropriately to satisfy whatever wildness arises. However, the purpose of an approximate solution to a complex problem is to convert chaos into order, not the reverse.Therefore, notice that the correspondent, perhaps inadvertently, raises an interesting point: in addition to the Universality Theorem, (see ([7], Chapter XI)), the function ξs (and hence, ζs) must also obey restraints embedded in the newly derived integral equations (61) and (66). Consider the consequence(s), taking note that the following results are all rigorously proven and widely accepted:(i)
The Universality Theorem recognizes that somewhere “further up the strip,” the functionζs will assume a “wild” value that approximates an arbitrary function within ϵ.(ii)
Equations (61) and (66) impose a strict relationship between ξs and ξ2iv on the s=0+2iv line, and therefore a relationship exists between ζs and ζ1−2iv through these equations (2) and the functional equation.(iii)
The functionζ1−2iv appearing in (61) and (66) via the functional equation and its polar-form is bounded by known limits on ζ1−2iv through well-established theorems. Crudely ζ1−2iv≪log2/32v, as well as other more strict limits given in [16] and [21] (see (131) and (132)).(iv)
The transfer functions embedded in (61) and (66) are simple polynomials whose maximum t−value anywhere on the contour is limited to Ot1, and these functions are the only elements in the integrand(s) that carry t dependence.(v)
The only other significant function appearing in the integrand of (61) and (66) is the product Γivπ−iv whose smoothness (continuity) and magnitude properties are well-known.(vi)
None of the elements of the integrand in (61) and (66) contain real poles or divergences and the integrals themselves are obviously convergent at both endpoints.(vii)
Consequently, the integrals in (61) and (66) are all limited in magnitude and constrained in their t−dependence and so cannot assume an arbitrarily “wild” value or approximate an arbitrarily “wild” function. This is a rigorous result contingent on the latter four results cited above, all of which are rigorous and true both in their own right as well as more general instances corresponding to any c>−1 (see Section 4).All of this suggests that an incompatibility exists among one or more of the above-cited rigorous results. Specifically, it is not apparent how an arbitrarily “wild” function, whose magnitude exceeds the known maximum value of the product of all the components of the integrands of (61) and (66) integrated over the range 0,∞, could be approximated “within ϵ” by ζσ+it, which is subject to the integral equation constraint(s) specified in the above. Furthermore, the correspondent’s observation that the approximate solutions (107) and (119) do not satisfy the Universality Theorem suggests that their precursor—the (rigorously proven) integral equation itself—does not satisfy the Universality Theorem. In summary, the reconciliation of the noted apparent incompatibility suggests a number of fruitful subjects for future investigation—again, beyond the scope of this work.


## 8.2. Lindelöf’s Hypothesis

### 8.2.1. A Second Correspondent Has Written

“(1) seems to indicate that t−1/4ζ1/2+it is unbounded as t⟶∞ which we know is not true; see [22]. In fact, the Lindelöf hypothesis states that ζ1/2+it=Otϵ as t⟶∞ for all ϵ>0, and if this is false, so is the Riemann hypothesis.”


### 8.2.2. The Author Comments

According to Edwards ([1], Section 9.2), Lindelöf’s theorem is based on Lindelöf’s observation that “there is a constant K such that(127)ζσ+it<Kt1/2−σ/2logt,throughout the half-strip 0≤ℜs≤1.” Assuming that ζ1/2+it asymptotically obeys a power law, Lindelöf denotes by μσ the least upper bound of the numbers A such that ζσ+itt−A is bounded as t⟶∞, shows that(128)0≤μ12≤14,and hypothesizes that μ1/2=0. This basically posits that as t⟶∞, ζ1/2+it increases at least faster than Ot0. Several proven refinements of (128) are known, over the years gradually reducing the upper inequality limit from 1/4 to the most recent [22] value 13/84=0.1548, yielding the least upper asymptotic bound(129)ζ12+it≪Ot13/84.By comparison, the equivalent result obtained here (see (119)) is(130)ζ12+itcosϕ≈4δ112,tcosΦtζ1−itt1/423/4π5/4,where cosϕ=−1k counts the zeros and −1≤cosΦt≤1. For emphasis, Φ has been written fully as Φt (see (15)). With ([21], Conjecture A)(131)maxt∈1,Tζ1+it∼expγlog logT,or ([16], Theorem 1)(132)maxt∈T,Tζ1+it≥expγlog logT+log log logT−C,both of which are valid for sufficiently large T, (130) becomes(133)ζ12+itcosϕ≈4δ112,tcosΦtt1/423/4π5/4loglogt+⋯.Notice the similarity between (107) (with (8)) and (127) for general σ, as well as (127) and (133) when σ=1/2. Up to this point, the functions δ1,2,3,4σ,t have been simply modelled as a function of σ, although clearly t dependence exists, e.g., see Figure 10(a). So, considering that δ11/2,t is expected to be a weak function of t as noted previously, it is reasonable to anticipate that t dependence residing in δ1σ,t and to a lesser extent cosΦt could affect any (dis)agreement between (127), (129), (133), and Lindelöf’s hypothesis.In contrast to Section8.1, where an arbitrary “wildness” was needed to generate agreement, here it is noted that the functions δ1,21/2,t capture the value of the integrand of the integral equation only in close proximity to the point v=t/2 in (105). Thus t-dependence of δ1,3 and δ2,4 is sensitive to the density of the real and imaginary zeros (respectively) of the function ζ1−2iv near a particular point on the 1-line. And, as previously noted, such t−dependence must persist for any c>−1, in which case it is useful to recall [19] that ℜζσ+it>0 if σ>1.192... corresponding to c>−0.904.... No arbitrariness here. In any case, it is evident that investigations into these observations might bear fruit. Again, this is beyond the scope of this work.


## 8.2.1. A Second Correspondent Has Written

“(1) seems to indicate that t−1/4ζ1/2+it is unbounded as t⟶∞ which we know is not true; see [22]. In fact, the Lindelöf hypothesis states that ζ1/2+it=Otϵ as t⟶∞ for all ϵ>0, and if this is false, so is the Riemann hypothesis.”


## 8.2.2. The Author Comments

According to Edwards ([1], Section 9.2), Lindelöf’s theorem is based on Lindelöf’s observation that “there is a constant K such that(127)ζσ+it<Kt1/2−σ/2logt,throughout the half-strip 0≤ℜs≤1.” Assuming that ζ1/2+it asymptotically obeys a power law, Lindelöf denotes by μσ the least upper bound of the numbers A such that ζσ+itt−A is bounded as t⟶∞, shows that(128)0≤μ12≤14,and hypothesizes that μ1/2=0. This basically posits that as t⟶∞, ζ1/2+it increases at least faster than Ot0. Several proven refinements of (128) are known, over the years gradually reducing the upper inequality limit from 1/4 to the most recent [22] value 13/84=0.1548, yielding the least upper asymptotic bound(129)ζ12+it≪Ot13/84.By comparison, the equivalent result obtained here (see (119)) is(130)ζ12+itcosϕ≈4δ112,tcosΦtζ1−itt1/423/4π5/4,where cosϕ=−1k counts the zeros and −1≤cosΦt≤1. For emphasis, Φ has been written fully as Φt (see (15)). With ([21], Conjecture A)(131)maxt∈1,Tζ1+it∼expγlog logT,or ([16], Theorem 1)(132)maxt∈T,Tζ1+it≥expγlog logT+log log logT−C,both of which are valid for sufficiently large T, (130) becomes(133)ζ12+itcosϕ≈4δ112,tcosΦtt1/423/4π5/4loglogt+⋯.Notice the similarity between (107) (with (8)) and (127) for general σ, as well as (127) and (133) when σ=1/2. Up to this point, the functions δ1,2,3,4σ,t have been simply modelled as a function of σ, although clearly t dependence exists, e.g., see Figure 10(a). So, considering that δ11/2,t is expected to be a weak function of t as noted previously, it is reasonable to anticipate that t dependence residing in δ1σ,t and to a lesser extent cosΦt could affect any (dis)agreement between (127), (129), (133), and Lindelöf’s hypothesis.In contrast to Section8.1, where an arbitrary “wildness” was needed to generate agreement, here it is noted that the functions δ1,21/2,t capture the value of the integrand of the integral equation only in close proximity to the point v=t/2 in (105). Thus t-dependence of δ1,3 and δ2,4 is sensitive to the density of the real and imaginary zeros (respectively) of the function ζ1−2iv near a particular point on the 1-line. And, as previously noted, such t−dependence must persist for any c>−1, in which case it is useful to recall [19] that ℜζσ+it>0 if σ>1.192... corresponding to c>−0.904.... No arbitrariness here. In any case, it is evident that investigations into these observations might bear fruit. Again, this is beyond the scope of this work.


## 9. Summary

Remark:.
throughout this work, it should be considered that the figures presented were all obtained from the computer program Maple [23] the accuracy of whose numerical evaluation of functions that do not lie in its mainstream is open to question. Spot check comparisons were performed between Figures from Maple and Mathematica [24], and no grounds for concern were found.
Although it is known in the literature thatξs satisfies integral equations as cited in Section 1, only the one presented by Patkowski ([4], Theorem (1.1)) comes close to sharing the desirable properties uncovered here. In particular, because the s−dependence is isolated within the transfer function, it became possible to associate properties of ξs anywhere in the critical strip with its properties on the 1-line, which are relatively well known. One surprising prediction is that the zeros of ξσ+it and those of ϒ1−it (and ϒit) are closely correlated, as demonstrated by a number of Figures. Although the correlation shown in the figures is not exact, the prediction is based on a model that leaves several questions unanswered:(i)
What is a more reasonable model to employ for the functionsδ1σ,t and δ4σ,t?(ii)
How will the two termsδ2σ,t and δ3σ,t affect (103) and (108) once they are incorporated into the model?(iii)
What will the numerical estimates look like at values oft significantly greater than those explored here, which are limited by restrictions on simple computational arithmetic?
These questions as well as those raised in Section8, implore further investigation.



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