# Around the Lipschitz Summation Formula

Authors: Wenbin Li; Hongyu Li; Jay Mehta
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Boundary behavior of important functions has been an object of intensive research since the time of Riemann. Kurokawa, Kurokawa-Koyama, and Chapman studied the boundary behavior of generalized Eisenstein series which falls into this category. The underlying principle is the use of the Lipschitz summation formula. Our purpose is to show that it is a form of the functional equation for the Lipschitz–Lerch transcendent (and in the long run, it is equivalent to that for the Riemann zeta-function) and that this being indeed a boundary function of the Hurwitz–Lerch zeta-function, one can extract essential information. We also elucidate the relation between Ramanujan’s formula and automorphy of Eisenstein series.


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## Body

## 1. Introduction

Boundary behavior of core functions has always been the object of intensive research since it exhibits a peculiar phenomenon that cannot be predicted by the behavior inside the domain. There are many instances of such unexpected behavior cf. [1–3]. Kurokawa [4] and Koyama and Kurokawa [5] studied the following limiting values by the Lipschitz summation formula:(1)limτ⟶xEk−1τ−τkEkτ,∀x∈ℝ,where Ekτ is the generalized Eisenstein series defined by (25).It has been elucidated and generalized by Chapman [6] who also used the Lipschitz summation formula for which he appealed to [7]. Knopp and Robbins in their Remarks 1 and 2 state their own views on the Lipschitz summation formula and Stark’s method [8] to the effect that they are not directly related to the functional equation (just as, for the Riemann zeta-function, the partial fraction expansion does not seem to be related). In [9], Murty and Sinha [10] result has been elucidated as a manifestation of one of the equivalent conditions to the functional equation, the Fourier–Bessel expansion, or the perturbed Dirichlet series ([11], Chapter 4), thereby explaining the genesis of Stark’s method. The Lipschitz summation formula for quadratic fields is also deduced there. We shall turn to this toward the end of Section 4.We cite the passage from [12] “The relation between modular forms and Dirichlet series with functional equations was discovered by Hecke, whose epoch-making work during the years 1930–1940, based on that discovery and that of the ‘Hecke operators’, brought out completely new aspects of a theory which many mathematicians would have regarded as a closed chapter long before.”We refer to this as part of the Riemann–Hecke–Bochner correspondence (RHB correspondence) ([11], p. 4 and 22) which is coined by Knopp [13].Our main aim in this paper is to prove the general modular relation, Theorem4, for the Lipschitz–Lerch transcendent (57) and deduce the general Lipschitz summation formula, Corollary 4. From this, we show that, in this case again, generalized RHB correspondence or the modular relation is the key for everything.But prior to this, in Section2, we state the modular relation for the Lambert series generated by the product of two Riemann zeta-functions with variables different by an odd integer and prove the automorphy of the Eisenstein series by the RHB correspondence, which of course settles the even weight case of (1). For another relation, cf. Bruinier and Funke [14].The even difference case turns out to be a reminiscent of the Wigert–Bellman divisor problem [15] as alluded to in [9]. In Section 3, we state the results in another form based on the shifted Mellin inversion.Here, we use a method similar to the one in [16] (pp. 73–75) of using the Ewald expansion ([11], Chapter 5) as opposed to the Fourier–Bessel expansion alluded to above. Since it is equivalent to the Lerch functional equation ([17], Theorem 5.3, p. 130) which in turn is equivalent to an asymmetric form (3) of the functional equation for the Riemann zeta-function, we thereby show that, in the long run, the genesis is in the functional equation for the Riemann zeta-function.As a necessary step, we show that the reciprocal Hurwitz formula amounts to a ramified functional equation, Lemma1. There are many cases of such ramified functional equations (cf. [18] and references therein). We state one of the earliest occurrences.In what follows, we always use the notations=σ+it as the complex variable.


## 2. An Example of the Riemann–Hecke–Bochner Correspondence

Throughout in what follows, we appeal to the Riemann zeta-function defined in the first instance forσ=Res>1 by(2)ζs=∏p1−1ps−1=∑n=1∞1ns.This satisfies the asymmetric form of the functional equation:(3)ζ1−s=22π−scosπ2sΓsζs,which is a prototype of the Hurwitz formula (cf. (64) for its reciprocal).We fix the integerα throughout.We consider the product of two zeta-functions:(4)φs=φs,α=ζsζs+α=∑n=1∞σ−αnns,where the series is absolutely convergent for σ>σφ≔max1,1−α, and(5)σ−αn=∑dnd−α,is the sum-of-divisors function. We note that φs−a includes the case of ζsζs−α as φs−a=∑n=1∞σαn/ns. This will be pursued in Section 3.The zeta-functionφs satisfies the asymmetric functional equation:(6)φ1−s=42π−2s+αcosπ2scosπ2s−αΓsΓs−αφs−α.Noting that(7)cosπ2s−α=sinπ2ssinπ2α,α=2ϰ+1,cosπ2scosπ2α,α=2ϰ,the product of cosines amounts to(8)12−1ϰsinπs,α=2ϰ+1,12−1ϰ1+cosπs,α=2ϰ.(i)
First we treat the case ofα=2ϰ+1 an odd integer. Then by the reciprocal relation for the gamma function, we see that the functional equation (6) amounts to(9)φ1−s=2−1ϰ2π−2s+απΓ1−sΓs−αφs−α.Now by the well-known procedure—Hecke gamma transform (e.g., [16]), we have for c>σφ and Rex>0(10)∑n=1∞σ−αne−nx=12πi∫cΓsφsx−sds,wherec indicates the Bromwich contour σ=c,−∞<t<∞.By a standard procedure of moving the line to the left up tod, where d<−α<0 (d=−α−1/2, say), whereby noting that the horizontal integrals vanish in the limit as t⟶∞, we obtain(11)12πi∫cΓsφsx−sds=12πi∫dΓsφsx−sds+Px,wherePx=Pαx is the residual function consisting of the sum of residues of the integrand at −a,…,−1,0,1,a+1. Writing 1−s for s in (11), we see that the right-hand side of (11) becomes 1/2πi∫1−dΓ1−sφ1−sxs−1ds+Px, whence substituting (9), we conclude that(12)∑n=1∞σ−αne−nx=−1ϰ2πα+11x12πi∫1−dΓs−αφs−α4π2x−sds+Px.Substituting the absolutely convergent series in (4) and factoring out x/4π2a, we deduce that(13)∑n=1∞σ−αne−nx=−1ϰ2πα+11xx4π2α∑n=1∞σ−αn×12πi∫1−dΓs−α4π2nx−s+αds+Px.Hence, by the Mellin inversion again, we obtain(14)∑n=1∞σ−αne−nx=−1α−1/2x2πα−1∑n=1∞σ−αne−4π2n/x+Px,forRex>0, i.e., the Bochner modular relation [19].Here only in the caseα=1>0, s=1 is a double pole, others being simple poles. In the case α=−1<0, there is one more term −1/2x−1.To compute the residual function, we use Table1, taking into account the trivial zeros of the Riemann zeta-function at negative even integers.For1≠α>0, the residual function is(15)Pαx=∑k=1α−1kk!ζ−kζ−k+αxk+ζ1+αx−1−12ζα=−∑j=1α/2+112j−1!ζ1−2jζα−2j+1x2j−1+ζ1+αx−1−12ζα,on writingk=2j−1. In literature, this is expressed in another form based on the explicit formula for zeta-values cf., e.g., ([20], p. 71 and 91):(16)ζ1−2k=−B2k2kk≥1,ζ2k=22k−12k!B2kπ2k,where the Bernoulli numbersB2k are b-notation ([20], p. 90).In particular, forα=2ϰ+1≥1, (14) with x replaced by 2πx amounts to the celebrated Ramanujan formula, cf., e.g., [21] for a general account:(17)∑n=1∞σ−2ϰ−1ne−2πnx+−1ϰ+1x2ϰ∑n=1∞σ−2ϰ−1ne−2πn/x=Px,wherePx=P2ϰ+1x is given by (15) whose concrete form is given in (49). Indeed, P2ϰ+1x is a residual function given as the sum of the residues:(18)Px=∑ξ∈RRess=ξ2π−sΓsζsζs+2ϰ+1x−s,whereR=−2ϰ−1,−2ϰ,−2ϰ+1,−2ϰ+3,…,−3,−1,0,1, and s=0 is a double pole only when ϰ=0 (others are simple poles). Hence,(19)P2ϰ+1x=∑j=1ϰ+1−1k2j+1!ζ1−2jζ2ϰ−2j+2x2j−1+ζ2ϰ+2x−1−12ζ2ϰ+1.cf. (49) below for another expression for the residual function.If we write−x=2πiτ, then τ∈ℋ and the Lambert series in Liouville’s form amounts to the Eisenstein series Ekτ in Definition 1, where k=a+1 is even and (14) gives(20)∑n=1∞σ−αne2πinτ=τα−1∑n=1∞σ−αne−2πin/τ+Px.In caseα=−2ϰ−1,ϰ≥1, (20) reads(21)∑n=1∞σ2ϰ+1ne2πinτ=τα−1∑n=1∞σ2ϰ+1ne−2πin/τ+Px,Pτ=P−2ϰ−1τ=2ϰ+1!ζ2ϰ+2−2πiτ−2ϰ−2−12ζ−2ϰ−1,cf. (51) and (52).In slightly different notation from [20] (p. 83),(22)G2kτ≔∑m,n∈ℤm,n≠0,01nτ+m2k.By Proposition 4 in [20] (p. 83), G2kτ is a modular form of weight 2k. The Laurent expansion (or q-expansion) reads ([20], p. 92)(23)G2kτ=2ζ2k+22πi2k2k−1!∑n=1∞σ2k−1ne2πinτ,τ∈ℋ.Appealing to (16), we have similarly to [20] (Corollary, p. 92)(24)12ζ2kG2kτ=2ζ1−2kE2kτ=−2kB2kE2kτ,where2/ζ1−2kE2kτ is the Eisenstein series ([20], (34), p. 92) and E2kτ is the even suffix case of the following.Table 1
Poles and the sum of residues in the relevant domain.
Case
σ
φ
d
Poles
P
α
x
α
>
0
1
d
=
α
−
1
/
2
−
α
,
…
,
−
1,0,1
(2.12)
α
<
0
1
−
α
d
=
−
1
/
2
0,1,1
−
α
Γ
1
−
α
ζ
1
−
α
x
α
−
1
−
1
/
2
ζ
αDefinition 1.
For anyk∈ℕ, Kurokawa introduces the general Eisenstein series:(25)Ekτ=ζ1−k2+∑n=1∞σk−1ne2πinτ,τ∈ℋ,which is not necessarily modular for k odd.
Stating (21) in explicit form(26)∑k=1∞σ2ϰ+1ke−2πkx−B2ϰ+24ϰ+4=−1ϰ+1x−2ϰ−2∑k=1∞σ2ϰ+1ke−2πk/x−B2ϰ+24ϰ+4,we see that it is nothing but(27)E2ϰ+2−1τ=τ2ϰ+2E2ϰ+2τ,i.e., the automorphy of E2ϰ+2z, cf. (25).
Thus, we have established.Theorem 1.
The Bochner modular relation (14) entails at one end of the spectrum α=−2ϰ+1 Ramanujan’s formula (17) and at the other end α=2ϰ+1 the automorphy of the Eisenstein series (27), thus abridging analytic number theory and the theory of modular forms.(ii)
Now we turn to the case ofa=2ϰ. We digress from (12) which should be replaced by(28)∑n=1∞σane−nx=2−1ϰ2π−2s+aπx12πi∫1−dπsinπsΓs−aφs4π2nx−sds+2−1ϰ2π−2s+aπx12πi∫1−dπcotπsΓs−aφs4π2nx−sds+Px.In the same way as we have deduced (13), we obtain(29)∑n=1∞σane−nx=2−1ϰ2π−2s+aπx12πi∫1−dπsinπsΓs−aφs4π2nx−sds+2−1ϰ2π−2s+aπx12πi∫1−dπcotπsΓs−aφs4π2nx−sds+Px.We shall stop here since it would be difficult to express the resulting integrals and Bellman’s method [15] yields an asymptotic formula rather than an equality. Partial theory of modular relations for the product of zeta-functions is given in ([11], Chapter 9, pp. 241–265), which is still in progress.Remark 1.
That the odd integer difference case (i) reduces to the one-gamma factor case to the RHB correspondence is not coincidental and is expounded in [11] (pp. 81–86), where one can also find a plausible discovery of Ramanujan of the transformation formula for the Dedekind eta-function η. Weil’s paper [22] is the most well-known paper that contains the proof of the latter, but prior to this, Chowla gave a proof [23] for the discriminant function, which is the 24th power of η. Ramanujan’s formula is stated as I, Entry 15 of Chapter 16 [24], Entry 21 (i), Chapter 14 of Ramanujan’s Notebook II [25] (which is and also as IV, Entry 20 of [26]). The most extensive account of information surrounding Ramanujan’s formula is [27], while [28] is the most informative account of special values of the zeta-functions. The intersection of references in these two excellent survey papers (which have a lot in common) is a null set. In literature, Ramanujan’s formula is stated for α=πx>0,β=π/x>0 satisfying the following relation:(30)αβ=π2,and in terms of Lambert series.
The Lambert seriesLz is defined for z|<1 by(31)Lz=∑n=1∞anzn1−zn,which is transformed into the Fourier series:(32)Lz=∑m=1∞∑n=1∞anzmn=∑ℓ=1∞bℓzℓ,the Liouville formula, where bℓ=∑d|ℓad. The sum-of-divisors function is the case an=nα. Original Ramanujan’s formula looks like having little to do with modular forms. Equation (17) being a rephrased Lambert series in Liouville’s form has amenity to the q-expansion, and so to automorphy.


## 3. The Ramanujan–Guinand Formula and Its Consequences

In Section2, we established that at both ends of the spectrum, the Bochner modular relation amounts to Ramanujan’s formula and the automorphy of Eisenstein series, respectively. In this section, we partially follow [29], reproduced in [11] (pp. 86–92), and elucidate the mechanism hidden in the correspondence φs⟷φs−a by differentiation of the Ramanujan–Guinand formula.In [14], they mention duality between the space of weak Maass forms of (negative) weight k∈1/2ℤ and the space of holomorphic cusp form of (positive) weight 2−k.To this end, we introduce theMellin inversion with shifted argumentIax. Let a≥0 be a fixed integer to be taken as the number of times of differentiation throughout. Let φs be the zeta-function defined by (4) with α=2ϰ+1 and ϰ a nonnegative integer:(33)φs=ζsζs+2ϰ+1,the other case being included in Theorem 2 below. The argument of its proof goes in the lines of Section 2. For Rex>0 let(34)Iax=Ia,ϰx=12πi∫c2π−sΓsφs−ax−sds,where c>1+a. The Hecke gamma transform reads(35)Iax=∑k=1∞σ−2ϰ−1kkae−2πkx.The special case,(36)Ix=I0x=12πi∫κ2π−sΓsφsx−sds=∑k=1∞σ−2ϰ−1ke−2πkx,is the Lambert series appearing in Ramanujan’s formula (17). Differentiating Ixa-times with respect to x, whereby we perform differentiation under integral sign, we have the additional factor(37)∏j=0a−1−s−j,which is −1aΓs+a/Γs, whence we deduce the remarkable formula:(38)dadxaIx=−2πaIax,i.e., a-times differentiation of the Lambert series (36) is effected by shifting the argument of φs by a in (36) and multiplying by −2πa. In view of (38), the a-times differentiated form of Ramanujan’s formula (17) amounts to a counterpart of the modular relation for Iax.Theorem 2.
(Ramanujan–Guinand formula). For the Mellin transformIax with shifted argument as defined by (34), we have the modular relation for ϰ≥0 and 0≤a≤2ϰ+1,(39)Iax=−1ϰ+a2π−ax−a+2ϰ∑k=0aak2ϰ−k!2ϰ−a!2πxkIk1x+Pax,where Pax is the residual function.Proof.
Proof depends on the following equation:(40)φs−a=−1n2π2ϰ+2s−2aΓa−2ϰ−sΓs−aφa−2ϰ−s,which is a variant of the functional equation (6) in the following form:(41)2π−sΓsφs=−1n2π2ϰ+sΓ−2ϰ−sφ−2ϰ−s.
Moving the line of integration toσ=−c1c1>2ϰ+1−a, we have(42)Iax=Jax+Pax,Jax=12πi∫−c1Γsφs−a2πx−sds,where Px=Pax denotes the sum of residues of the integrand at its poles at s=a−2ϰ−1,a−2ϰ,a−2ϰ+1,a−2ϰ+3,…,0,a+1.
Substitute (40) and change the variable s⟷a−2ϰ−s in the integral Jax.
Then(43)Jax=−1ϰ2πi∫a−2ϰ+c1Γa−2ϰ−sΓsΓ−2ϰ−sφs2π−s−axs−a+2ϰds.
Substituting(44)Γa−2ϰ−sΓ−2ϰ−s=Γ1+s+2ϰΓ1+s+2ϰ−asinπssinπs−a=−1aΓa+2ϰ+1Γs+2ϰ+1−a,we find that(45)Jax=−1ϰ+a2πi∫a−2ϰ+c1Γs+2nϰ+1ΓsΓs+2ϰ+1−aφs2π−s−axs−a+2ϰds.
Since the gamma factor can be computed as follows for0≤a≤2ϰ+1,(46)ΓsΓs+2ϰ+1Γs+2ϰ+1−a=∑k=0aak2ϰ−k!2ϰ−a!Γs+k,where for a=2ϰ+1, the right-hand side is to mean Γs+2ϰ+1, we conclude from (42) and (45) that(47)Iax=−1ϰ+a2π−ax−a+2ϰ∑k=0aak2ϰ−k!2ϰ−a!×12πi∫a−2ϰ+c1Γs+kφs2πx−sds+Pax,where the sum reduces to 1 for a−2ϰ+1. Finally, we note that the integral on the right-hand side of (47) becomes by the change of variable s⟷s+k(48)2πxkIk1x.
Hence, (47) leads to (39), completing the proof.Corollary 1.
(i)
The casea=0 is Ramanujan’s formula (17) with the residual function
(49)
P
x
=
2
π
2
ϰ
+
1
2
x
∑
j
=
0
ϰ
+
1
−
1
j
B
2
j
2
j
!
B
2
ϰ
+
2
−
2
j
2
ϰ
+
2
−
2
j
!
x
2
ϰ
+
2
−
2
j
+
−
1
2
ζ
2
ϰ
+
1
1
+
−
1
ϰ
+
1
x
2
ϰ
if
ϰ
≥
1
,
1
2
log
x
if
ϰ
=
0
,
valid forRex>0.
(ii)
The casea=2ϰ is Guinand’s formula, cf. ([29], Theorem 3) with
(50)
P
2
ϰ
x
=
2
ϰ
!
ζ
2
ϰ
+
2
2
π
x
−
2
ϰ
−
1
+
1
2
ζ
−
2
ϰ
−
1
2
π
x
+
−
1
ϰ
2
2
π
−
2
ϰ
2
ϰ
!
ζ
2
ϰ
+
1
,
ϰ
≥
1.
(iii)
The casea=2ϰ+1 reads
(51)
I
2
ϰ
+
1
x
=
−
1
ϰ
+
1
2
π
−
2
ϰ
−
1
x
−
2
ϰ
−
2
I
2
ϰ
+
1
1
x
+
P
2
ϰ
+
1
x
,
where
(52)
P
2
ϰ
+
1
x
=
2
ϰ
+
1
!
ζ
2
ϰ
+
2
2
π
x
−
2
ϰ
−
2
−
1
2
ζ
−
2
ϰ
−
1
ϰ
≥
1
.
These lead to (21), thence to the automorphy (27).
(iv)
The special case of (50) with ϰ=1, i.e., once differentiated form of Ramanujan’s formula, yields Terras’ formula [30, 31]:
(53)
ζ
3
=
2
45
π
3
−
4
∑
k
=
1
∞
e
−
2
π
k
σ
−
3
k
2
π
2
k
2
+
π
k
+
1
2
.Remark 2.
The Mellin inversion with shifted argument (34) is an additive version of the “pseudomodular relation principle,” which is a processed modular relation with the processing gamma factor Γs+a ([11], p. 50). In this special case, the Main Formula (38) is the manifestation of the statement of Razar that the differentiation of Lambert series essentially corresponds to the shift of the argument of the associated Dirichlet series.Remark 3.
It is shown ([11], pp. 82–84) that the special case ϰ=0,a=0 of Theorem 2, the Ramanujan–Guinand formula, is nothing other than the automorphic property of the Dedekind eta-function, cf. Remark 1 above. This can be regarded as once differentiated form of Guinand’s formula. We note that, in the case of ϰ=0, the twice differentiated Guinand’s formula, with a suitable modification, coincides with the automorphic property of the Eisenstein series E2:(54)E2τ+3πy=1−24∑k=1∞kqk1−qk=12πilogητ′.
Equation (51), once differentiated form of Guinand’s formula (=2n+1 times differentiated form of Ramanujan’s formula=the negative case thereof) leads to (27), automorphy of G2ϰ+2.


## 4. Lipschitz Summation Formula

Knopp–Robbins [7] in their Remark 1 state their view on the Lipschitz summation formula to the effect that it is conceptually simpler than Riemann’s original method of using the theta series. However, at least the special case of the Lipschitz summation formula (Theorem 3) which is applied by Chapman to establish the limit relation has already been used extensively and can be readily deduced from the partial fraction expansion for the cotangent function. Since it is known that the partial fraction expansion is equivalent to the functional equation for the Riemann zeta-function, we may say that Chapman’s result is a consequence of the functional equation. By Corollary 4 below, we shall show that the Lipschitz summation formula itself is equivalent to the functional equation, thereby enhancing the above statement. Pasles and Pribitkin [32] extend the Lipschitz summation formula to the two-variable case to which we hope to return elsewhere.Theorem 3 (Lipschitz summation formula).
For the complex variablesz=x+iy,x>0, s=σ+it,σ>1 and the real parameter 0<α≤1, we have the Lipschitz summation formula:(55)2πsΓs∑m=0∞m+αs−1e−2πzm+α=∑n=−∞∞e2πinαz+ins.Under the condition0<α<1, this formula holds in the wider half-plane σ>0.In what follows, we shall generalize Theorem3 in a wider framework, deducing Corollary 4 to Theorem 4 as a general modular relation.Let(56)Φw,s,z=∑n=0∞wnn+zs,be the Hurwitz–Lerch zeta-function, and let(57)ϕx,s,z=Φe2πinx,s,x=∑n=0∞e2πinxn+zs,denote the boundary function—the Lipschitz–Lerch transcendent (([9], pp. 59–62), ([2], pp. 128–131), [33]). This is in close correspondence with the case of Lsz and ℓsx considered in [34]. As a consequence of the main formula in Appendix Section, we may deduce, from the reciprocal Hurwitz formula (64), a generalized Lipschitz summation formula (78) which is indeed a form of the functional equation.Lemma 1.
The Dirichlet series(58)φs=1πs/2ℓsx=∑n=1∞e2πixnπns,(59)ψ1s=12πs/2ζs,x+12πs/2ζs,1−x=∑n=1∞1/2πn+x−1s+1/2πn−xs,(60)ψ2s=−i2πs/2ζs,x+i2πs/2ζs,1−x=∑n=1∞−i/2πn+x−1s+i/2πn−xs,satisfy the ramified functional equation, a special case of (A.2) with r=1(61)Γs2Γ12+s2φs=Γ12−s2Γ12+s2ψ11−s+Γ−s2Γ1+s2ψ21−s,or(62)χs=Γs−0,12,12,12;;−−φs,Γ1−s0,120,12;;−−ψ11−s+Γ1−s−12,12−12,12;;−−ψ21−s.Proof.
From Euler’s formula, we have(63)es/2i1−s=sins2s+icoss2s,e−s/2i1−s=sins2s−icoss2s.
We transform the reciprocal Hurwitz formula0<x<1:(64)ℓsx=Γ1−s2π1−se1−s2πiζ1−s,x+e−1−s/2πiζ1−s,1−x=f1sf2s,say, where f1s=Γ1−s/2π1−s is the first factor of the right-hand side member and f2s is the second. Substituting (63), the second factor becomes(65)f2s=sinπ2sζ1−s,x+ζ1−s,1−x+cosπ2sζ1−s,x−ζ1−s,1−x.
In view of this, we transform the first factor accordingly(66)f1s=πs−3/22Γ1−s/2Γs/2πsinπ/2sor=πs−3/22Γ1−s/2Γ1+s/2πcosπ/2s.
Substituting (65) and (66), we transform (64) into(67)ℓsx=πs−1/2Γ1/2−s/22Γs/2ζ1−s,x−ζ1−s,1−x+iπs−1/2Γ1−s/22Γs/2+1/2ζ1−s,x−ζ1−s,1−x=π−s/2Γ1/2−s/22Γs/2ψ1s−π−s/2Γ1−s/22Γs/2+1/2ψ2s,where ψi’s are defined in (59) and (60). Clearing the denominators in (67),(68)φs=Γs2Γ12+s21πs/2lsx=Γ12−s2Γ12+s2ψ1s−Γs2Γ1−s2ψ2s.
It remains to note that the factor−Γs/2Γ1−s/2 of the second summand may be transformed into Γ−s/2Γ1+s/2, which is indeed the case in view of the reciprocity formula.
By Lemma1, we have the ramified functional equation. The only pole of χs which is neither a pole of Γ1/2+s/2 nor of Γ1+s/2 is s1=0. Thus, by main formula (A.13), we have the following modular relation. For the notation and formulas, we refer to ([11], pp. 49–59).Theorem 4.
(69)
1
π
s
/
2
∑
n
=
1
∞
e
2
π
i
n
x
n
s
H
z
π
n
Δ
⊕
−
;
−
s
/
2
,
1
/
2
,
1
/
2
+
s
/
2
,
1
/
2
;
−
=
1
2
π
1
−
s
/
2
∑
n
=
1
∞
1
n
+
x
−
1
1
−
s
H
π
n
+
x
−
1
z
Δ
∗
⊕
1
−
s
/
2
,
1
/
2
;
−
1
−
s
/
2
,
1
/
2
;
−
+
1
n
−
x
1
−
s
H
π
n
−
x
z
Δ
∗
⊕
1
−
s
/
2
,
1
/
2
;
−
1
−
s
/
2
,
1
/
2
;
−
+
1
2
π
1
−
s
/
2
∑
k
=
1
∞
−
i
n
+
x
−
1
1
−
s
H
π
n
+
x
−
1
z
Δ
∗
⊕
−
s
/
2
,
1
/
2
;
−
−
s
/
2
,
1
/
2
;
−
+
i
n
−
x
1
−
s
H
π
n
−
x
z
Δ
∗
⊕
−
s
/
2
,
1
/
2
;
−
−
s
/
2
,
1
/
2
;
−
+
Res
Γ
w
−
s
Δ
χ
w
z
s
−
w
,
w
=
0
=
1
2
π
1
−
s
/
2
∑
n
=
1
∞
1
n
+
x
−
1
1
−
s
H
π
n
+
x
−
1
z
Δ
∗
⊕
1
−
s
/
2
,
1
/
2
;
−
1
−
s
/
2
,
1
/
2
;
−
−
i
H
π
k
+
a
−
1
z
Δ
∗
⊕
−
s
/
2
,
1
/
2
;
−
−
s
/
2
,
1
/
2
;
−
+
1
n
−
x
1
−
s
H
π
n
−
x
z
Δ
∗
⊕
1
−
s
/
2
,
1
/
2
;
−
1
−
s
/
2
,
1
/
2
;
−
+
i
H
π
n
−
x
z
Δ
∗
⊕
−
s
/
2
,
1
/
2
;
−
−
s
/
2
,
1
/
2
;
−
+
Res
Γ
w
−
s
Δ
χ
w
z
s
−
w
,
w
=
0
.Corollary 2.
IfΔ=1,1−;;−−, we have Δ∗=−0,1;;−−, and so Theorem 4 amounts to(70)1πs/2∑n=1∞e2πinansH2,11,2zπn1,1s2,12,12+s2,12=12π1−s/2∑n=1∞1n+a−11−sH1,22,1πn+a−1z1−s2,120,1,1−s2,12−iH1,22,1πn+a−1z−s2,120,1,−s2,12+1n−a1−sH1,22,1πn−az1−s2,120,1,1−s2,12+iH1,22,1πn−az−s2,120,1,−s2,12+ResΓs−wχwzs−w,w=0.Corollary 3.
(Ewald expansion).(71)ϕx,s,z=∑n=1∞e2πinxn+zs+1zs=∑n=1∞e−2πn+x−1zi2πn+x−1e−πi/21−sΓ1−s,−2πn+x−1zi+e2πn−xzi2πn−xeπi/21−sΓ1−s,2πn−xzi+12zs,where Γs,c is the incomplete gamma function defined as follows:(72)Γs,c=Γs−cs∫01e−cuus−1du.Proof.
We transformH-functions in Corollary 2 in a concrete form. The procedure is the same for the three H-functions, i.e., duplication formula, H⟶G formula, and the explicit formula for the G-function, and we use the known results on them freely, cf. [11, 35], etc. We have(73)H2,11,2z1,1s2,12,12+s/2,12=21−sπG1,11,12z1s=21−sπΓs2zs1+2zs,(74)H1,22,1z12−s2,120,1,12−s2,12=12πH1,33,1z212−s2,120,12,12,12,12−s2,12=1πG1,33,1z241−s20,12,1−s2=Γse1−s/2πi−ziΓ1−s,−zi+e−1−s/2πi+ziΓ1−s,zi,and similarly to (74)(75)H1,22,1z−s2,120,1,−s2,12=Γs+1e−s/2πi−ziΓ−s,−zi+es/2πi+ziΓ−s,zi=Γsie1−s/2πi−ziΓ1−s,−zi−ie−1−s/2πi+ziΓ1−s,zi+2zs.Hence, (71) amounts to(76)12πs−1∑n=1∞e2πinxn+ξs=1π1−s/2∑n=1∞e1−s/2πie−2πn+x−1ξin+x−11−sΓ1−s,−2πn+x−1ξi+e−1−s/2πie2πn−xξin−x1−sΓ1−s,2πn−xξi−i2πξsn+x−1+i2πξsn−x+12πs−1ξsℓ0x,where ξ=1/2πz. Since(77)ℓ0x=−12+i2π1x+∑n=1∞1x+n+1x−n,equation (76) leads to the incomplete gamma series (71) for (57).
We are now in a position to transform (71) into the functional equation.Corollary 4.
(General Lipschitz summation formula)(78)ϕx,s,z=Γ1−s2π1−se1−s/2πi−2πxziϕ−z,1−s,x+e−1−s/2πi+2π1−xziϕz,1−s,1−x,where 0<z,x<1, and t=−zu−1.Proof.
By (72), we have the left-hand side of (71) is equal to(79)=Γ1−s∑k=1∞e−2πn+x−1zi2πn+x−1e−πi/21−s+e2πn−xzi2πn−xeπi/21−s−1zs−1∑k=1∞∫01e2πn+x−1ziu−1+e−2πn−xziu−1u−sdu+12zs=Γ1−s2π1−se1−s/2πi−2πxziϕ−z,1−s,x+e−1−s/2πi+2π1−zziϕz,1−s,1−x−1zs∑k∈ℤ∫0ze2πn−xti1−tz−sdt+12zs.
Equation (78) is sometimes referred to as the Lipschitz summation formula ([36–38] etc.).
The character analogue of the Lipschitz summation formula is known ([2], pp. 128–131), and so we may naturally treat a more general case which will be conducted elsewhere.
We briefly state the Lipschitz summation formula for quadratic fields which is contained in ([11], Chapter 4). This is an elucidation of Koshlyakov’s results [39–42]. Let Ω be a quadratic field whose degree ϰ=r1+r2=2 with its discriminant Δ. Let(80)A=2r2πϰ/2Δ,and let r=r1+r2−1 denote the rank of the unit group. Let ζΩs be the Dedekind zeta-function of Ω with the coefficients αk=βk=αk, with αk indicating the number of ideals of norm k, where λk=Ak, μk=Ak, r=1, and(81)ρ=2r+1πr2RhwΔ=−2r+1πr2ζΩrsΔ.
The functional equation reads(82)Γr112sΓr2sφs=Γr112−12sΓr21−sφ1−s,where(83)φs=A−sζΩs=∑n=1∞αnAns.
In conformity with Koshlyakov ([40], p. 241) (cf. (83)), we introduce the perturbed Dedekind zeta-function:(84)ζΩs,z=−2ζΩ0w+∑n=1∞αnn+zs.
Then the Fourier–Bessel expansion gives the Lipschitz summation formula for an imaginary quadratic field ([11], (4.42)):(85)∑n=−∞∞′αnz+ins=eπi/2sζΩs,iz+e−πi/2sζΩs,−iz=Asz1/21−sΓs∑n=1∞αnn1/21−sεs+1K1−s2Aεnz+ε¯s+1K1−s2Aε¯nz−ζΩ0zs,which is the corrected form of (23.15) in [40] , where ε=eπ/4i and the prime on the summation sign means that the term with n=0 is excluded.Theorem 5 (Theorem 4.6 in [11]).
The generating Dirichlet series for the Wigert–Bellman divisor problem, cf. (29) above,(86)ζΩs,z=z−sΓs∑n=1∞αnn12πi∫ctanπ2wΓ1−wΓs−w1nz−wdw,amounts to the Lipschitz summation formula(87)12∑n=−∞∞αnz+ins=12eπi/2sζΩs,iz+12e−πi/2sζΩs,−iz=Asz1/21−sΓs∑n=1∞αnn1/21−sεs+1iK1−s2Aεnz−ε¯s+1iK1−s2Aε¯nz.This is the corrected form of (23.16) in [40].


## 5. Ramified Functional Equations

There are some instances of the ramified functional equations in literature.In [43–45], they are stated in the case of zeta-functions with periodic coefficients which satisfy the ramified functional equations as a result of representations in bases consisting of the Hurwirz and Lerch zeta-functions [18]: suppose fn be a periodic function with period M,(88)Ls,f=∑n=1∞fnns,be the associated Dirichlet series absolutely convergent σ=Res>1 and that(89)fodd=12fnmodM−f−nmodM,feven=12fnmodM+f−nmodM,be odd, resp. even part of f: f=feven+fodd. Then(90)L1−s,f=πM1/2−sΓs/2Γ1−s/2Ls,f^even+Γ1+s/2Γ1−s/2Ls,f^odd,which amounts to (61) on clearing the denominators and multiplying by π/Ms−1/2. Wang and Banerjee [46] treat the product of Hurwitz zeta-functions which satisfy a ramified functional equation as a result of the Hurwitz formula:(91)ζ1−s,x=Γs2πse−πis/2ℓsx+eπis/2ℓs1−x,whose reciprocal (64) has been used and proved to be a ramified functional equation in the proof of Lemma 1 above. More general case is that of the Hurwitz–Lerch zeta-function described in [35] (pp. 27–31) and [33] (pp. 121–126, 339–341). Another class of zeta-functions that satisfy a ramified functional equations is that of Barnes multiple zeta-functions cf. [33] (pp. 77–88) and [47]. For their rich applications [48] and references therein. Hardy and Littlewood [49, 50] use the Barnes double zeta-function.In another context, Estermann [51] and others and in Prehomogeneous Vector Space (PHV) theory [52].



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