# A Rademacher Type Formula for  Partitions and Overpartitions

Authors: Andrew V. Sills
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions ofn and the Zuckerman formula for the Fourier coefficients of ϑ4(0∣τ)-1 is presented.


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## Body

## 1. Background

### 1.1. Partitions

Apartition of an integer n is a representation of n as a sum of positive integers, where the order of the summands (called parts) is considered irrelevant. It is customary to write the parts in nonincreasing order. For example, there are three partitions of the integer 3, namely, 3, 2+1, and 1+1+1. Let p(n) denote the number of partitions of n, with the convention that p(0)=1, and let f(x) denote the generating function of p(n), that is, let(1.1)f(x):=∑n=0∞p(n)xn.Euler [1] was the first to systematically study partitions. He showed that(1.2)f(x)=∏m=1∞11-xm.
Euler also showed that(1.3)1f(x)=∑n=-∞∞(-1)nxn(3n-1)/2,
and since the exponents appearing on the right side of (1.3) are the pentagonal numbers, (1.3) is often called “Euler's pentagonal number theorem."Although Euler's results can all be treated from the point of view of formal power series, the series and infinite products above (and indeed all the series and infinite products mentioned in this paper) converge absolutely when|x|<1, which is important for analytic study of these series and products.Hardy and Ramanujan were the first to studyp(n) analytically and produced an incredibly accurate asymptotic formula [2, page 85, equation (1.74)], namely,(1.4)p(n)=12π2∑k=1⌊αn⌋k∑0≦h<k(h,k)=1ω(h,k)e-2πihn/kddn(exp((π/k)(2/3)(n-1/24))n-1/24)+O(n-1/4),
where(1.5)ω(h,k)=exp(πi∑r=1k-1rk(hrk-⌊hrk⌋-12)).α is an arbitrary constant, and here and throughout (h,k) is an abbreviation for gcd(h,k).Later Rademacher [3] improved upon (1.4) by finding the following convergent series representation for p(n):(1.6)p(n)=1π2∑k=1∞k∑0≦h<k(h,k)=1ω(h,k)e-2πinh/kddn(sinh((π/k)(2/3)(n-1/24))n-1/24).Rademacher's method was used extensively by many practitioners, including Grosswald [4, 5], Haberzetle [6], Hagis [7–15], Hua [16], Iseki [17–19], Lehner [20], Livingood [21], Niven [22], and Subramanyasastri [23] to study various restricted partitions functions.Recently, Bringmann and Ono [24] have given exact formulas for the coefficients of all harmonic Maass forms of weight ≦1/2. The generating functions considered herein are weakly holomorphic modular forms of weight -1/2, and thus they are harmonic Maass forms of weight ≦1/2. Accordingly, the results of this present paper could be derived from the general theorem in [24]. However, here we opt to derive the results via classical method of Rademacher.


### 1.2. Overpartitions

Overpartitions were introduced by Corteel and Lovejoy in [25] and have been studied extensively by them and others including Bringmann, Chen, Fu, Goh, Hirschhorn, Hitczenko, Lascoux, Mahlburg, Robbins, Rødseth, Sellers, Yee, and Zhao [25–43].Anoverpartition of n is a representation of n as a sum of positive integers with summands in nonincreasing order, where the last occurrence of a given summand may or may not be overlined. Thus the eight overpartitions of 3 are 3, 3¯, 2+1, 2¯+1, 2+1¯, 2¯+1¯, 1+1+1, 1+1+1¯.Letp¯(n) denote the number of overpartitions of n and let f¯(x) denote the generating function ∑n=0∞p¯(n)xn of p¯(n). Elementary techniques are sufficient to show that(1.7)f¯(x)=∏m=1∞1+xm1-xm=f(x)2f(x2).Note that(1.8)1f¯(x)=∑n=-∞∞(-1)nxn2.
via an identity of Gauss (see equation (2.2.12), page 23 in [44, 45]), so that the reciprocal of the generating function for overpartitions is a series wherein a coefficient is nonzero if and only if the exponent of x is a perfect square, just as the reciprocal of the generating function for partitions is a series wherein a coefficient is nonzero if and only if the exponent of x is a pentagonal number.Hardy and Ramanujan, writing more than 80 years before the coining of the term “overpartition," stated [2, page 109-110] that the function which we are calling p¯(n) “has no very simple arithmetical interpretation; but the series is none the less, as the direct reciprocal of a simple ϑ-function, of particular interest." They went on to state that(1.9)p¯(n)=14πddn(eπnn)+32πcos(23nπ-16π)ddn(eπn/3)+⋯+O(n-1/4).In fact, (1.9) was improved to the following Rademacher-type convergent series by Zuckerman [46, page 321, equation (8.53)]:(1.10)p¯(n)=12π∑k≧12∤kk∑0≦h<k(h,k)=1ω(h,k)2ω(2h,k)e-2πinh/kddn(sinh(πn/k)n).
A simplified proof of (1.10) was given by Goldberg in his Ph.D. thesis [47].


### 1.3. Partitions Where No Odd Part Is Repeated

Letpod(n) denote the number of partitions of n where no odd part appears more than once. Let g(x) denote the generating function of pod(n), so we have(1.11)g(x)=∑n=0∞pod(n)xn=∏m=1∞1+x2m-11-x2m=f(x)f(x4)f(x2).Via another identity of Gauss (see equation (2.2.13), page 23 in [44, 45]), it turns out that(1.12)1g(x)=∑n=0∞(-x)n(n+1)/2=∑n=-∞∞(-1)nx2n2-n;
so in this case the reciprocal of the generating function under consideration has nonzero coefficients at the exponents which are triangular (or equivalently, hexagonal) numbers.The analogous Rademacher-type formula forpod(n) is as follows:(1.13)pod(n)=2π∑k≧1k(1-(-1)k+⌊(k,4)4⌋)×∑0≦h<k(h,k)=1ω(h,k)ω(4h/(k,4),k/(k,4))ω(2h/(k,2),k/(k,2))e-2πinh/k×ddn(sinh(π(k,4)(8n-1)/4k)8n-1).Equation (1.13) is the case r=2 of Theorem 2.1 to be presented in the next section.


## 1.1. Partitions

Apartition of an integer n is a representation of n as a sum of positive integers, where the order of the summands (called parts) is considered irrelevant. It is customary to write the parts in nonincreasing order. For example, there are three partitions of the integer 3, namely, 3, 2+1, and 1+1+1. Let p(n) denote the number of partitions of n, with the convention that p(0)=1, and let f(x) denote the generating function of p(n), that is, let(1.1)f(x):=∑n=0∞p(n)xn.Euler [1] was the first to systematically study partitions. He showed that(1.2)f(x)=∏m=1∞11-xm.
Euler also showed that(1.3)1f(x)=∑n=-∞∞(-1)nxn(3n-1)/2,
and since the exponents appearing on the right side of (1.3) are the pentagonal numbers, (1.3) is often called “Euler's pentagonal number theorem."Although Euler's results can all be treated from the point of view of formal power series, the series and infinite products above (and indeed all the series and infinite products mentioned in this paper) converge absolutely when|x|<1, which is important for analytic study of these series and products.Hardy and Ramanujan were the first to studyp(n) analytically and produced an incredibly accurate asymptotic formula [2, page 85, equation (1.74)], namely,(1.4)p(n)=12π2∑k=1⌊αn⌋k∑0≦h<k(h,k)=1ω(h,k)e-2πihn/kddn(exp((π/k)(2/3)(n-1/24))n-1/24)+O(n-1/4),
where(1.5)ω(h,k)=exp(πi∑r=1k-1rk(hrk-⌊hrk⌋-12)).α is an arbitrary constant, and here and throughout (h,k) is an abbreviation for gcd(h,k).Later Rademacher [3] improved upon (1.4) by finding the following convergent series representation for p(n):(1.6)p(n)=1π2∑k=1∞k∑0≦h<k(h,k)=1ω(h,k)e-2πinh/kddn(sinh((π/k)(2/3)(n-1/24))n-1/24).Rademacher's method was used extensively by many practitioners, including Grosswald [4, 5], Haberzetle [6], Hagis [7–15], Hua [16], Iseki [17–19], Lehner [20], Livingood [21], Niven [22], and Subramanyasastri [23] to study various restricted partitions functions.Recently, Bringmann and Ono [24] have given exact formulas for the coefficients of all harmonic Maass forms of weight ≦1/2. The generating functions considered herein are weakly holomorphic modular forms of weight -1/2, and thus they are harmonic Maass forms of weight ≦1/2. Accordingly, the results of this present paper could be derived from the general theorem in [24]. However, here we opt to derive the results via classical method of Rademacher.


## 1.2. Overpartitions

Overpartitions were introduced by Corteel and Lovejoy in [25] and have been studied extensively by them and others including Bringmann, Chen, Fu, Goh, Hirschhorn, Hitczenko, Lascoux, Mahlburg, Robbins, Rødseth, Sellers, Yee, and Zhao [25–43].Anoverpartition of n is a representation of n as a sum of positive integers with summands in nonincreasing order, where the last occurrence of a given summand may or may not be overlined. Thus the eight overpartitions of 3 are 3, 3¯, 2+1, 2¯+1, 2+1¯, 2¯+1¯, 1+1+1, 1+1+1¯.Letp¯(n) denote the number of overpartitions of n and let f¯(x) denote the generating function ∑n=0∞p¯(n)xn of p¯(n). Elementary techniques are sufficient to show that(1.7)f¯(x)=∏m=1∞1+xm1-xm=f(x)2f(x2).Note that(1.8)1f¯(x)=∑n=-∞∞(-1)nxn2.
via an identity of Gauss (see equation (2.2.12), page 23 in [44, 45]), so that the reciprocal of the generating function for overpartitions is a series wherein a coefficient is nonzero if and only if the exponent of x is a perfect square, just as the reciprocal of the generating function for partitions is a series wherein a coefficient is nonzero if and only if the exponent of x is a pentagonal number.Hardy and Ramanujan, writing more than 80 years before the coining of the term “overpartition," stated [2, page 109-110] that the function which we are calling p¯(n) “has no very simple arithmetical interpretation; but the series is none the less, as the direct reciprocal of a simple ϑ-function, of particular interest." They went on to state that(1.9)p¯(n)=14πddn(eπnn)+32πcos(23nπ-16π)ddn(eπn/3)+⋯+O(n-1/4).In fact, (1.9) was improved to the following Rademacher-type convergent series by Zuckerman [46, page 321, equation (8.53)]:(1.10)p¯(n)=12π∑k≧12∤kk∑0≦h<k(h,k)=1ω(h,k)2ω(2h,k)e-2πinh/kddn(sinh(πn/k)n).
A simplified proof of (1.10) was given by Goldberg in his Ph.D. thesis [47].


## 1.3. Partitions Where No Odd Part Is Repeated

Letpod(n) denote the number of partitions of n where no odd part appears more than once. Let g(x) denote the generating function of pod(n), so we have(1.11)g(x)=∑n=0∞pod(n)xn=∏m=1∞1+x2m-11-x2m=f(x)f(x4)f(x2).Via another identity of Gauss (see equation (2.2.13), page 23 in [44, 45]), it turns out that(1.12)1g(x)=∑n=0∞(-x)n(n+1)/2=∑n=-∞∞(-1)nx2n2-n;
so in this case the reciprocal of the generating function under consideration has nonzero coefficients at the exponents which are triangular (or equivalently, hexagonal) numbers.The analogous Rademacher-type formula forpod(n) is as follows:(1.13)pod(n)=2π∑k≧1k(1-(-1)k+⌊(k,4)4⌋)×∑0≦h<k(h,k)=1ω(h,k)ω(4h/(k,4),k/(k,4))ω(2h/(k,2),k/(k,2))e-2πinh/k×ddn(sinh(π(k,4)(8n-1)/4k)8n-1).Equation (1.13) is the case r=2 of Theorem 2.1 to be presented in the next section.


## 2. A Common Generalization

Let us define(2.1)fr(x):=f(x)f(x2r)f(x2),
where r is a nonnegative integer. Thus,(2.2)f0(x)=f¯(x)=∑n=0∞p¯(n)xn,f1(x)=f(x)=∑n=0∞p(n)xn,f2(x)=g(x)=∑n=0∞pod(n)xn.Letpr(n) denote the coefficient of xn in the expansion of fr(n), that is,(2.3)fr(x)=∑n=0∞pr(n)xn.Notice thatfr(x) can be represented by several forms of equivalent infinite products, each of which has a natural combinatorial interpretation:
(2.4)fr(x)=∏m=1∞1+xm1-x2rm(2.5)=∏m=1∞1(1-x2m-1)(1-x2rm)(2.6)=∏m=1∞11-x2r-1m∏λ=12r-1-1(1+x2r-1m+λ).Thus,pr(n) equals each of the following:(i)
the number of overpartitions ofn where nonoverlined parts are multiples of 2r (by (2.4));(ii)
the number of partitions ofn where all parts are either odd or multiples of 2r (by (2.5)), provided r≧1;(iii)
the number of partitions ofn where nonmultiples of 2r-1 are distinct (by (2.6)), provided r≧1.Theorem 2.1.
Forr=0,1,2,3,4,
(2.7)pr(n)=2(r+1)/23π∑k≧1k(k,2max(r,1))=1×∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)×ddn{sinh(π(24n-2r+1)(1+2r-1)/(2r/2·6k))24n-2r+1}+3π∑j=1+⌊r/2⌋r2(2-j+r)/2∑k≧1k(k,2r)=2j×∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)×ddn{sinh(π(24n-2r+1)(-1+22j-r)/6k)24n-2r+1}.


## 3. A Proof of Theorem2.1

The method of proof is based on Rademacher's proof of (1.6) in [48] with the necessary modifications. Additional details of Rademacher's proof of (1.6) are provided in [49], [50, Chapter 14], and [51, Chapter 5].Of fundamental importance is the path of integration to be used. In [48], Rademacher improved upon his original proof of (1.6) given in [3], by altering his path of integration from a carefully chosen circle to a more complicated path based on Ford circles, which in turn led to considerable simplifications later in the proof.


### 3.1. Farey Fractions

The sequenceℱN of proper Farey fractions of orderN is the set of all h/k with (h,k)=1 and 0≦h/k<1, arranged in increasing order. Thus, for example, ℱ4={0/1,1/4,1/3,1/2,2/3,3/4}.For a givenN, let hp, hs, kp, and ks be such that hp/kp is the immediate predecessor of h/k and hs/ks is the immediate successor of h/k in ℱN. It will be convenient to view each ℱN cyclically, that is, to view 0/1 as the immediate successor of (N-1)/N.


### 3.2. Ford Circles and the Rademacher Path

Leth and k be integers with (h,k)=1 and 0≦h<k. The Ford circle [52] C(h,k) is the circle in ℂ of radius 1/2k2 centered at the point(3.1)hk+12k2i.Theupper  arcγ(h,k)of the Ford circleC(h,k) is those points of C(h,k) from the initial point(3.2)αI(h,k):=hk-kpk(k2+kp2)+1k2+kp2i
to the terminal point(3.3)αT(h,k):=hk+ksk(k2+ks2)+1k2+ks2i,
traversed in the clockwise direction.Note that we have(3.4)αI(0,1)=αT(N-1,N).Every Ford circle is in the upper half plane. Forh1/k1,h2/k2∈ℱN, C(h1,k1) and C(h2,k2) are either tangent or do not intersect.TheRademacher pathP(N) of order N is the path in the upper half of the τ-plane from i to i+1 consisting of(3.5)⋃h/k∈ℱNγ(h,k)
traversed left to right and clockwise. In particular, we consider the left half of the Ford circle C(0,1) and the corresponding upper arc γ(0,1) to be translated to the right by 1 unit. This is legal given then periodicity of the function which is to be integrated over P(N).


### 3.3. Set Up the Integral

Letn and r be fixed, with n>(2r-1)/24.Since(3.6)fr(x)=∑n=0∞pr(n)xn,Cauchy's residue theorem implies that(3.7)pr(n)=12πi∫𝒞fr(x)xn+1dx,
where 𝒞 is any simply closed contour enclosing the origin and inside the unit circle. We introduce the change of variable(3.8)x=e2πiτ
so that the unit disk |x|≦1 in the x-plane maps to the infinitely tall, unit-wide strip in the τ-plane where 0≦ℜτ≦1 and ℑτ≧0. The contour 𝒞 is then taken to be the preimage of P(N) under the map x↦e2πiτ.Better yet, let us replacex with e2πiτ in (3.7) to express the integration in the τ-plane:(3.9)pr(n)=∫P(N)fr(e2πiτ)e-2πinτdτ=∑h/k∈ℱN∫γ(h,k)fr(e2πiτ)e-2πinτdτ=∑k=1N∑0≦h<k(h,k)=1∫γ(h,k)fr(e2πiτ)e-2πinτdτ.


### 3.4. Another Change of Variable

Next, we change variables again, taking(3.10)τ=iz+hk,
so that(3.11)z=-ik(τ-hk).
Thus C(h,k) (in the τ-plane) maps to the clockwise-oriented circle Kk(-) (in the z-plane) centered at 1/2k with radius 1/2k.So we now have(3.12)pr(n)=i∑k=1Nk-1∑0≦h<k(h,k)=1e-2πinh/k∫zI(h,k)zT(h,k)arcofKk(-)e2nπz/kfr(e2πih/k-2πz/k)dz,
where zI(h,k) (resp., zT(h,k)) is the image of αI(h,k) (see (3.2)) (resp. αT(h,k) [see 3.2]) under the transformation (3.11).So the transformation (3.11) maps the upper arc γ(h,k) of C(h,k) in the τ-plane to the arc on Kk(-) which initiates at(3.13)zI(h,k)=kk2+kp2+kpk2+kp2i
and terminates at(3.14)zT(h,k)=kk2+ks2-ksk2+ks2i.


### 3.5. Exploiting a Modular Transformation

From the theory of modular forms, we have the transformation formula [2, page 93, Lemma 4.31]:(3.15)f(exp(2πihk-2πzk))=ω(h,k)exp(π(z-1-z)12k)zf(exp(2πiiz-1+Hk)),
where z is the principal branch, (h,k)=1, and H is a solution to the congruence(3.16)hH≡-1(modk).
From (3.15), we deduce the analogous transformation for fr(x).The transformation formula is a piecewise defined function withr+1 cases corresponding to j=0,1,2,…,r, where (k,2r)=2j:(3.17)fr(exp(2πihk-2πzk))=ω(h,k)ω(2r-jh,2-jk)ω(2h/(2-δj0),k/(2-δj0))exp(π(2+22j-r+1-(2-δj0)2)24kz+π(1-2r)z12k)×z2r-j-1(2-δj0)×f(exp(-2π/kz+2Hjπi/k))f(exp(-22j-r+1π/kz+22j-r+1Hjπi/k))f(exp(-π(2-δj0)2/kz+Hjπ(2-δj0)2i/k)),
where Hj is divisible by 2r-j and is a solution to the congruence hHj≡-1(modk), and
(3.18)δj0={1ifj=0,0ifj≠0
is the Kronecker δ-function.Notice that in particular, for⌊r/2⌋≦j≦r, (3.17) simplifies to
(3.19)fr(exp(2πihk-2πzk))=ω(h,k)ω(2r-jh,2-jk)ω(h,k/2)exp(π12k[(22j-r-1)z-1+(1-2r)z])×z2r-jf2j-r(exp(-2πkz+2Hjπik)).Since ther=0 case was established by Zuckerman, and the r=1 case by Rademacher, we will proceed with the assumption that r>1.Apply (3.17) to (3.12) to obtain
(3.20)pr(n)=i∑j=0r∑k=1N(k,2r)=jk-1∑0≦h<k(h,k)=1e-2πinh/k×∫zI(h,k)zT(h,k)arcofKk(-)ω(h,k)ω(2r-jh,2-jk)ω(2h/(2-δj0),k/(2-δj0))×exp(π(2+22j-r+1-(2-δj0)2)24kz+π(24n+1-2r)z12k)×z2r-j-1(2-δj0)×f(exp(-2π/kz+2Hjπi/k))f(exp(-22j-r+1π/kz+22j-r+1Hjπi/k))f(exp(-π(2-δj0)2/kz+Hjπ(2-δj0)2i/k))dz.


### 3.6. Normalization

Next, introduce a normalizationζ=zk (this is not strictly necessary, but it will allow us in the sequel to quote various useful results directly from the literature):(3.21)pr(n)=i∑j=0r∑k=1N(k,2r)=jk-5/2∑0≦h<k(h,k)=1e-2πinh/k×∫ζI(h,k)ζT(h,k)arcofK(-)ω(h,k)ω(2r-jh,2-jk)ω(2h/(2-δj0),k/(2-δj0))×exp(π(2+22j-r+1-(2-δj0)2)24ζ+π(24n+1-2r)ζ12k2)×ζ2r-j-1(2-δj0)×f(exp(-2π/ζ+2Hjπi/k))f(exp(-22j-r+1π/ζ+22j-r+1Hjπi/k))f(exp(-π(2-δj0)2/ζ+Hjπ(2-δj0)2i/k))dζ,
where
(3.22)ζI(h,k)=k2k2+kp2+kkpk2+kp2i,ζT(h,k)=k2k2+ks2-kksk2+ks2i.Let us now rewrite (3.21) as
(3.23)pr(n)=i∑j=0r∑k=1N(k,2max(r,1))=2jk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(2h/(2-δj0),k/(2-δj0))×(ℐj,1+ℐj,2),
where
(3.24)ℐj,1:=∫ζI(h,k)ζT(h,k)arcexp(π(2+22j-r+1-(2-δj0)2)24ζ+π(24n+1-2r)ζ12k2)×ζ2r-j-1(2-δj0)×{-1+f(exp(-2π/ζ+2Hjπi/k))f(exp(-22j-r+1π/ζ+22j-r+1Hjπi/k))f(exp(-π(2-δj0)2/ζ+Hjπ(2-δj0)2i/k))}dζ,ℐj,2:=∫ζI(h,k)ζT(h,k)arcexp(π(2+22j-r+1-(2-δj0)2)24ζ+π(24n+1-2r)ζ12k2)×ζ2r-j-1(2-δj0)dζ.


### 3.7. Estimation

It will turn out that asN→∞, only ℐj,2 for j=0 and ⌊r/2⌋<j≦r ultimately make a contribution. Note that all the integrations in the ζ-plane occur on arcs and chords of the circle K of radius 1/2 centered at the point 1/2. So, inside and on K, 0<ℜζ≦1 and ℜ(1/ζ)≧1.


#### 3.7.1. Estimation ofℐj,2 for 1≦j≦⌊r/2⌋

The regularity of the integrand allows us to alter the path of integration from the arc connectingζI(h,k) and ζT(h,k) to the directed segment. By [51, page 104, Theorem 5.9], the length of the path of integration does not exceed 22k/N, and on the segment connecting ζI(h,k) to ζT(h,k), |ζ|<2k/N. Thus, the absolute value of the integrand is(3.25)|exp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)ζ2r-j|=|ζ|1/22(r-j)/2exp((24n+1-2r)πℜζ12k2)exp(π(22j-r-1)12ℜ1ζ)≦|ζ|1/22r/2exp(2πn).Thus, for1≦j≦⌊r/2⌋,(3.26)|ℐj,2|≦22kN(2kN)1/22r/2e2πn≦Cj′k3/2N-3/2for a constantCj′  (recalling that n and r are fixed).


#### 3.7.2. Estimation ofℐj,1 for 1≦j≦⌊r/2⌋

We have the absolute value of the integrand:(3.27)|ζ2r-jexp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)|×|-1+f(exp(-2π/ζ+2Hjπi/k))f(exp(-22j-r+1π/ζ+22j-r+1Hjπi/k))f(exp(-4πζ+4Hjπik))|=|ζ2r-jexp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)|×|-1+f2j-r(exp(-2πζ+2Hjπik))|=|ζ|1/22(r-j)/2exp((24n+1-2r)πℜζ12k2)exp((22j-r-1)π12ℜ1ζ)×|∑m=1∞p2j-r(m)exp(-2πmζ+2Hjπimk)|≦|ζ|1/22(r-j)/2exp((24n+1-2r)π12)∑m=1∞p2j-r(m)exp(-π12(24m-22j-r+1))≦|ζ|1/22r/2e2πn∑m=1∞p0(m)e-2πm=cj|ζ|1/2for a constantcj. So, for 1≦j≦⌊r/2⌋,(3.28)|ℐj,1|≦22kN(2kN)1/2cj<Cjk3/2N-3/2for a constantCj.


#### 3.7.3. Estimation ofI0,1

Letpr*(x) be defined by(3.29)∑n=0∞pr*(n)xn=f(x2r)f(x)f(x2r-1).Again, the regularity of the integrand allows us to alter the path of integration from the arc connectingζI(h,k) and ζT(h,k) to the directed segment.With this in mind, we estimate the absolute value of the integrand:(3.30)|exp(π(1+21-r)24ζ+π(24n+1-2r)ζ12k2)ζ2r-1|×|-1+f(exp(-2π/ζ+2H0πi/k))f(exp(-21-rπ/ζ+21-rH0πi/k))f(exp(-π/ζ+H0πi/k))|=|exp(π(1+21-r)24ζ+π(24n+1-2r)ζ12k2)ζ2r-1|×|∑m=1∞p*(m)exp(-21-rπmζ+21-rH0πimk)|=exp(π(1+21-r)24ℜ1ζ)exp(π(24n+1-2r)ℜζ12k2)|ζ|1/22(r-1)/2×|∑m=1∞p*(m)exp(-21-rπmζ)exp(21-rH0πimk)|≦e2πn|ζ|1/22(r-1)/2×∑m=1∞|p*(m)|exp(π(1+21-r)24ℜ1ζ-21-rπmℜ1ζ)=e2πn|ζ|1/22(r-1)/2∑m=1∞|p*(m)|exp(-π24·2r-1ℜ1ζ(24m-1-2r-1))≦e2πn|ζ|1/22(r-1)/2∑m=1∞|p*(m)|exp(-π24(24m-1-2r-1))<e2πn|ζ|1/22(r-1)/2∑m=1∞|p*(24m-1-2r-1)|y24m-1-2r-1(wherey=e-π/24)=c0|ζ|1/2for a constantc0. So,(3.31)|ℐ0,1|≦22kN(2kN)1/2c0<C0k3/2N-3/2for a constantC0.


#### 3.7.4. Estimation ofℐj,1 for 1+⌊r/2⌋≦j≦r≦4

Again, the regularity of the integrand allows us to alter the path of integration from the arc connectingζI(h,k) and ζT(h,k) to the directed segment.With this in mind,(3.32)|exp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)ζ2r-j|×|-1+f2j-r(exp(-2πζ+2Hjπik))|=|exp(π(22j-r-1)12ζ)exp(π(24n+1-2r)ζ12k2)ζ2r-j|×|∑m=1∞p2j-r(m)exp(-2πmζ)exp(2Hjπimk)|=exp(π(22j-r-1)12ℜ1ζ)exp(π(24n+1-2r)ℜζ12k2)|ζ|1/22(r-j)/2×|∑m=1∞p2j-r(m)exp(-2πmℜ1ζ)exp(2Hjπimk)|≦e2πn|ζ|1/22r/2∑m=1∞p2j-r(m)exp(-π12ℜ1ζ(24m-22j-r+1))≦e2πn|ζ|1/22r/2∑m=1∞p0(m)exp(-π12ℜ1ζ(24m-2r+1))=cj|ζ|1/2
for a constant cj. So,(3.33)|ℐj,1|≦22kN(2kN)1/2cj<Cjk3/2N-3/2for a constantCj, when 1+⌊r/2⌋≦j≦r.


#### 3.7.5. Combining the Estimates

One has(3.34)|i∑k=1N(k,2r)=1k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)ℐ0,1+i∑j=1⌊r/2⌋∑k=1N(k,2r)=2jk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)(ℐj,1+ℐj,2)+i∑j=1+⌊r/2⌋r∑k=1N(k,2r)=2jk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)ℐj,1|<∑j=0r∑k=1N∑h=0k-1Cjk-1N-3/2+∑j=1⌊r/2⌋∑k=1N∑h=0k-1C'jk-1N-3/2≦C′′N-32∑k=1N1,(whereC′′=∑j=0rCj+∑j=1⌊r/2⌋Cj′)=O(N-1/2).
Thus, we may revise (3.21) to(3.35)pr(n)=i∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)ℐ0,2+i∑j=1+⌊r/2⌋r∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)ℐj,2+O(N-1/2).


### 3.8. Evaluation ofℐj,2 for j=0 and 1+⌊r/2⌋≦j≦r

Writeℐj,2 as(3.36)ℐj,2=∫K(-)exp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)ζ2r-jdζ-ℐj,3-ℐj,4,
where(3.37)ℐj,3:=∫0ζI(h,k),ℐj,4:=∫ζT(h,k)0,
and ℐj,3 and ℐj,4 have the same integrand as (3.36) (and analogously for I0,2).


#### 3.8.1. Estimation ofℐj,3 and ℐj,4

We note that the length of the arc of integration inℐj,3 is less than πk/2N, and on this arc|ζ|<2k/N. [50, page 272]. Also, ℜ1/ζ=1 on K [50, page 271, equation (120.2)]. Further, 0<ℜζ<2k2/N2 [50, page 271, equation (119.6)]. The absolute value of the integrand is thus
(3.38)|2r-jζ|1/2exp((24n+1-2r)πℜζ12k2+(22j-r-1)π12ℜ1ζ)<2(r-j)/221/4k1/2N-1/2exp((24n+1-2r)π6N2+(22j-r-1)π12)
so that(3.39)|ℐj,2|<πk2-1/2N-12(r-j)/221/4k1/2N-1/2exp((24n+1-2r)π6N2+(22j-r-1)π12)=πk3/2N-3/22(2r-2j-1)/4exp((24n+1-2r)π6N2+(22j-r-1)π12)=O(k3/2N-3/2exp((24n+1-2r)π6N2)).By the same reasoning,|ℐj,3|=O(k3/2N-3/2exp((24n+1-2r)π/6N2)).We may therefore revise (3.35) to
(3.40)pr(n)=i∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)×∫K(-)ζ2r-1exp{π(21-r+1)24ζ+π(24n+1-2r)ζ12k2}dζ+i∑j=1+⌊r/2⌋r∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)×∫K(-)ζ2r-jexp{π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2}dζ+O(N-1/2)
and upon letting N tend to infinity, obtain(3.41)pr(n)=i∑k=1∞k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)×∫K(-)ζ2r-1exp{π(21-r+1)24ζ+π(24n+1-2r)ζ12k2}dζ+i∑j=1+⌊r/2⌋r∑k=1∞k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)×∫K(-)ζ2r-jexp{π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2}dζ.


### 3.9. The Final Form

We may now introduce the change of variable(3.42)ζ=π(2+22j-r+1-(2-δj0)2)24t
(where the first summation in (3.41) is the j=0 term separated out for clarity), which allows the integral to be evaluated in terms of I3/2, the Bessel function of the first kind of order 3/2 with purely imaginary argument [53, page 372, §17.7] when we bear in mind that a “bent” path of integration is allowable according to the remark preceding Equation (8) on page 177 of [54]. See also [51, page 109]. The final form of the formula is then obtained by using the fact that Bessel functions of half-odd integer order can be expressed in terms of elementary functions.We therefore have, forc=π(2+22j-r+1-(δj0)2)/24,(3.43)pr(n)=i∑k=1∞k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)2(r-1)/2(π21-r+124)3/2×∫c-i∞c+i∞t-5/2exp{t+(1+21-r)(24n+1-2r)π2288k2t}dt+i∑j=1+⌊r/2⌋r∑k=1∞k-5/2×∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)×2(r-j)/2(π22j-r-112)3/2×∫c-i∞c+i∞t-5/2exp{t+(22j-r-1)(24n-2r+1)π2144k2t}dt=π(24n-2r+1)3/4×{(1+2r-1)3/42(r-2)/4∑k≧1(k,2max(r,1))=1)k-1×∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)I3/2(π(24n-2r+1)(1+2r-1)2r/2·6k)+∑j=1+⌊r/2⌋r(22j-r-1)3/42(2-j+r)/2∑k≧1(k,2r)=2jk-1×∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)I3/2(π(24n-2r+1)(1+22j-r)6k)},
which, after application of the formula [51, page 110]
(3.44)I3/2(z)=2zπddz(sinhzz),
is equivalent to Theorem 2.1.


## 3.1. Farey Fractions

The sequenceℱN of proper Farey fractions of orderN is the set of all h/k with (h,k)=1 and 0≦h/k<1, arranged in increasing order. Thus, for example, ℱ4={0/1,1/4,1/3,1/2,2/3,3/4}.For a givenN, let hp, hs, kp, and ks be such that hp/kp is the immediate predecessor of h/k and hs/ks is the immediate successor of h/k in ℱN. It will be convenient to view each ℱN cyclically, that is, to view 0/1 as the immediate successor of (N-1)/N.


## 3.2. Ford Circles and the Rademacher Path

Leth and k be integers with (h,k)=1 and 0≦h<k. The Ford circle [52] C(h,k) is the circle in ℂ of radius 1/2k2 centered at the point(3.1)hk+12k2i.Theupper  arcγ(h,k)of the Ford circleC(h,k) is those points of C(h,k) from the initial point(3.2)αI(h,k):=hk-kpk(k2+kp2)+1k2+kp2i
to the terminal point(3.3)αT(h,k):=hk+ksk(k2+ks2)+1k2+ks2i,
traversed in the clockwise direction.Note that we have(3.4)αI(0,1)=αT(N-1,N).Every Ford circle is in the upper half plane. Forh1/k1,h2/k2∈ℱN, C(h1,k1) and C(h2,k2) are either tangent or do not intersect.TheRademacher pathP(N) of order N is the path in the upper half of the τ-plane from i to i+1 consisting of(3.5)⋃h/k∈ℱNγ(h,k)
traversed left to right and clockwise. In particular, we consider the left half of the Ford circle C(0,1) and the corresponding upper arc γ(0,1) to be translated to the right by 1 unit. This is legal given then periodicity of the function which is to be integrated over P(N).


## 3.3. Set Up the Integral

Letn and r be fixed, with n>(2r-1)/24.Since(3.6)fr(x)=∑n=0∞pr(n)xn,Cauchy's residue theorem implies that(3.7)pr(n)=12πi∫𝒞fr(x)xn+1dx,
where 𝒞 is any simply closed contour enclosing the origin and inside the unit circle. We introduce the change of variable(3.8)x=e2πiτ
so that the unit disk |x|≦1 in the x-plane maps to the infinitely tall, unit-wide strip in the τ-plane where 0≦ℜτ≦1 and ℑτ≧0. The contour 𝒞 is then taken to be the preimage of P(N) under the map x↦e2πiτ.Better yet, let us replacex with e2πiτ in (3.7) to express the integration in the τ-plane:(3.9)pr(n)=∫P(N)fr(e2πiτ)e-2πinτdτ=∑h/k∈ℱN∫γ(h,k)fr(e2πiτ)e-2πinτdτ=∑k=1N∑0≦h<k(h,k)=1∫γ(h,k)fr(e2πiτ)e-2πinτdτ.


## 3.4. Another Change of Variable

Next, we change variables again, taking(3.10)τ=iz+hk,
so that(3.11)z=-ik(τ-hk).
Thus C(h,k) (in the τ-plane) maps to the clockwise-oriented circle Kk(-) (in the z-plane) centered at 1/2k with radius 1/2k.So we now have(3.12)pr(n)=i∑k=1Nk-1∑0≦h<k(h,k)=1e-2πinh/k∫zI(h,k)zT(h,k)arcofKk(-)e2nπz/kfr(e2πih/k-2πz/k)dz,
where zI(h,k) (resp., zT(h,k)) is the image of αI(h,k) (see (3.2)) (resp. αT(h,k) [see 3.2]) under the transformation (3.11).So the transformation (3.11) maps the upper arc γ(h,k) of C(h,k) in the τ-plane to the arc on Kk(-) which initiates at(3.13)zI(h,k)=kk2+kp2+kpk2+kp2i
and terminates at(3.14)zT(h,k)=kk2+ks2-ksk2+ks2i.


## 3.5. Exploiting a Modular Transformation

From the theory of modular forms, we have the transformation formula [2, page 93, Lemma 4.31]:(3.15)f(exp(2πihk-2πzk))=ω(h,k)exp(π(z-1-z)12k)zf(exp(2πiiz-1+Hk)),
where z is the principal branch, (h,k)=1, and H is a solution to the congruence(3.16)hH≡-1(modk).
From (3.15), we deduce the analogous transformation for fr(x).The transformation formula is a piecewise defined function withr+1 cases corresponding to j=0,1,2,…,r, where (k,2r)=2j:(3.17)fr(exp(2πihk-2πzk))=ω(h,k)ω(2r-jh,2-jk)ω(2h/(2-δj0),k/(2-δj0))exp(π(2+22j-r+1-(2-δj0)2)24kz+π(1-2r)z12k)×z2r-j-1(2-δj0)×f(exp(-2π/kz+2Hjπi/k))f(exp(-22j-r+1π/kz+22j-r+1Hjπi/k))f(exp(-π(2-δj0)2/kz+Hjπ(2-δj0)2i/k)),
where Hj is divisible by 2r-j and is a solution to the congruence hHj≡-1(modk), and
(3.18)δj0={1ifj=0,0ifj≠0
is the Kronecker δ-function.Notice that in particular, for⌊r/2⌋≦j≦r, (3.17) simplifies to
(3.19)fr(exp(2πihk-2πzk))=ω(h,k)ω(2r-jh,2-jk)ω(h,k/2)exp(π12k[(22j-r-1)z-1+(1-2r)z])×z2r-jf2j-r(exp(-2πkz+2Hjπik)).Since ther=0 case was established by Zuckerman, and the r=1 case by Rademacher, we will proceed with the assumption that r>1.Apply (3.17) to (3.12) to obtain
(3.20)pr(n)=i∑j=0r∑k=1N(k,2r)=jk-1∑0≦h<k(h,k)=1e-2πinh/k×∫zI(h,k)zT(h,k)arcofKk(-)ω(h,k)ω(2r-jh,2-jk)ω(2h/(2-δj0),k/(2-δj0))×exp(π(2+22j-r+1-(2-δj0)2)24kz+π(24n+1-2r)z12k)×z2r-j-1(2-δj0)×f(exp(-2π/kz+2Hjπi/k))f(exp(-22j-r+1π/kz+22j-r+1Hjπi/k))f(exp(-π(2-δj0)2/kz+Hjπ(2-δj0)2i/k))dz.


## 3.6. Normalization

Next, introduce a normalizationζ=zk (this is not strictly necessary, but it will allow us in the sequel to quote various useful results directly from the literature):(3.21)pr(n)=i∑j=0r∑k=1N(k,2r)=jk-5/2∑0≦h<k(h,k)=1e-2πinh/k×∫ζI(h,k)ζT(h,k)arcofK(-)ω(h,k)ω(2r-jh,2-jk)ω(2h/(2-δj0),k/(2-δj0))×exp(π(2+22j-r+1-(2-δj0)2)24ζ+π(24n+1-2r)ζ12k2)×ζ2r-j-1(2-δj0)×f(exp(-2π/ζ+2Hjπi/k))f(exp(-22j-r+1π/ζ+22j-r+1Hjπi/k))f(exp(-π(2-δj0)2/ζ+Hjπ(2-δj0)2i/k))dζ,
where
(3.22)ζI(h,k)=k2k2+kp2+kkpk2+kp2i,ζT(h,k)=k2k2+ks2-kksk2+ks2i.Let us now rewrite (3.21) as
(3.23)pr(n)=i∑j=0r∑k=1N(k,2max(r,1))=2jk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(2h/(2-δj0),k/(2-δj0))×(ℐj,1+ℐj,2),
where
(3.24)ℐj,1:=∫ζI(h,k)ζT(h,k)arcexp(π(2+22j-r+1-(2-δj0)2)24ζ+π(24n+1-2r)ζ12k2)×ζ2r-j-1(2-δj0)×{-1+f(exp(-2π/ζ+2Hjπi/k))f(exp(-22j-r+1π/ζ+22j-r+1Hjπi/k))f(exp(-π(2-δj0)2/ζ+Hjπ(2-δj0)2i/k))}dζ,ℐj,2:=∫ζI(h,k)ζT(h,k)arcexp(π(2+22j-r+1-(2-δj0)2)24ζ+π(24n+1-2r)ζ12k2)×ζ2r-j-1(2-δj0)dζ.


## 3.7. Estimation

It will turn out that asN→∞, only ℐj,2 for j=0 and ⌊r/2⌋<j≦r ultimately make a contribution. Note that all the integrations in the ζ-plane occur on arcs and chords of the circle K of radius 1/2 centered at the point 1/2. So, inside and on K, 0<ℜζ≦1 and ℜ(1/ζ)≧1.


### 3.7.1. Estimation ofℐj,2 for 1≦j≦⌊r/2⌋

The regularity of the integrand allows us to alter the path of integration from the arc connectingζI(h,k) and ζT(h,k) to the directed segment. By [51, page 104, Theorem 5.9], the length of the path of integration does not exceed 22k/N, and on the segment connecting ζI(h,k) to ζT(h,k), |ζ|<2k/N. Thus, the absolute value of the integrand is(3.25)|exp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)ζ2r-j|=|ζ|1/22(r-j)/2exp((24n+1-2r)πℜζ12k2)exp(π(22j-r-1)12ℜ1ζ)≦|ζ|1/22r/2exp(2πn).Thus, for1≦j≦⌊r/2⌋,(3.26)|ℐj,2|≦22kN(2kN)1/22r/2e2πn≦Cj′k3/2N-3/2for a constantCj′  (recalling that n and r are fixed).


### 3.7.2. Estimation ofℐj,1 for 1≦j≦⌊r/2⌋

We have the absolute value of the integrand:(3.27)|ζ2r-jexp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)|×|-1+f(exp(-2π/ζ+2Hjπi/k))f(exp(-22j-r+1π/ζ+22j-r+1Hjπi/k))f(exp(-4πζ+4Hjπik))|=|ζ2r-jexp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)|×|-1+f2j-r(exp(-2πζ+2Hjπik))|=|ζ|1/22(r-j)/2exp((24n+1-2r)πℜζ12k2)exp((22j-r-1)π12ℜ1ζ)×|∑m=1∞p2j-r(m)exp(-2πmζ+2Hjπimk)|≦|ζ|1/22(r-j)/2exp((24n+1-2r)π12)∑m=1∞p2j-r(m)exp(-π12(24m-22j-r+1))≦|ζ|1/22r/2e2πn∑m=1∞p0(m)e-2πm=cj|ζ|1/2for a constantcj. So, for 1≦j≦⌊r/2⌋,(3.28)|ℐj,1|≦22kN(2kN)1/2cj<Cjk3/2N-3/2for a constantCj.


### 3.7.3. Estimation ofI0,1

Letpr*(x) be defined by(3.29)∑n=0∞pr*(n)xn=f(x2r)f(x)f(x2r-1).Again, the regularity of the integrand allows us to alter the path of integration from the arc connectingζI(h,k) and ζT(h,k) to the directed segment.With this in mind, we estimate the absolute value of the integrand:(3.30)|exp(π(1+21-r)24ζ+π(24n+1-2r)ζ12k2)ζ2r-1|×|-1+f(exp(-2π/ζ+2H0πi/k))f(exp(-21-rπ/ζ+21-rH0πi/k))f(exp(-π/ζ+H0πi/k))|=|exp(π(1+21-r)24ζ+π(24n+1-2r)ζ12k2)ζ2r-1|×|∑m=1∞p*(m)exp(-21-rπmζ+21-rH0πimk)|=exp(π(1+21-r)24ℜ1ζ)exp(π(24n+1-2r)ℜζ12k2)|ζ|1/22(r-1)/2×|∑m=1∞p*(m)exp(-21-rπmζ)exp(21-rH0πimk)|≦e2πn|ζ|1/22(r-1)/2×∑m=1∞|p*(m)|exp(π(1+21-r)24ℜ1ζ-21-rπmℜ1ζ)=e2πn|ζ|1/22(r-1)/2∑m=1∞|p*(m)|exp(-π24·2r-1ℜ1ζ(24m-1-2r-1))≦e2πn|ζ|1/22(r-1)/2∑m=1∞|p*(m)|exp(-π24(24m-1-2r-1))<e2πn|ζ|1/22(r-1)/2∑m=1∞|p*(24m-1-2r-1)|y24m-1-2r-1(wherey=e-π/24)=c0|ζ|1/2for a constantc0. So,(3.31)|ℐ0,1|≦22kN(2kN)1/2c0<C0k3/2N-3/2for a constantC0.


### 3.7.4. Estimation ofℐj,1 for 1+⌊r/2⌋≦j≦r≦4

Again, the regularity of the integrand allows us to alter the path of integration from the arc connectingζI(h,k) and ζT(h,k) to the directed segment.With this in mind,(3.32)|exp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)ζ2r-j|×|-1+f2j-r(exp(-2πζ+2Hjπik))|=|exp(π(22j-r-1)12ζ)exp(π(24n+1-2r)ζ12k2)ζ2r-j|×|∑m=1∞p2j-r(m)exp(-2πmζ)exp(2Hjπimk)|=exp(π(22j-r-1)12ℜ1ζ)exp(π(24n+1-2r)ℜζ12k2)|ζ|1/22(r-j)/2×|∑m=1∞p2j-r(m)exp(-2πmℜ1ζ)exp(2Hjπimk)|≦e2πn|ζ|1/22r/2∑m=1∞p2j-r(m)exp(-π12ℜ1ζ(24m-22j-r+1))≦e2πn|ζ|1/22r/2∑m=1∞p0(m)exp(-π12ℜ1ζ(24m-2r+1))=cj|ζ|1/2
for a constant cj. So,(3.33)|ℐj,1|≦22kN(2kN)1/2cj<Cjk3/2N-3/2for a constantCj, when 1+⌊r/2⌋≦j≦r.


### 3.7.5. Combining the Estimates

One has(3.34)|i∑k=1N(k,2r)=1k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)ℐ0,1+i∑j=1⌊r/2⌋∑k=1N(k,2r)=2jk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)(ℐj,1+ℐj,2)+i∑j=1+⌊r/2⌋r∑k=1N(k,2r)=2jk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)ℐj,1|<∑j=0r∑k=1N∑h=0k-1Cjk-1N-3/2+∑j=1⌊r/2⌋∑k=1N∑h=0k-1C'jk-1N-3/2≦C′′N-32∑k=1N1,(whereC′′=∑j=0rCj+∑j=1⌊r/2⌋Cj′)=O(N-1/2).
Thus, we may revise (3.21) to(3.35)pr(n)=i∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)ℐ0,2+i∑j=1+⌊r/2⌋r∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)ℐj,2+O(N-1/2).


## 3.7.1. Estimation ofℐj,2 for 1≦j≦⌊r/2⌋

The regularity of the integrand allows us to alter the path of integration from the arc connectingζI(h,k) and ζT(h,k) to the directed segment. By [51, page 104, Theorem 5.9], the length of the path of integration does not exceed 22k/N, and on the segment connecting ζI(h,k) to ζT(h,k), |ζ|<2k/N. Thus, the absolute value of the integrand is(3.25)|exp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)ζ2r-j|=|ζ|1/22(r-j)/2exp((24n+1-2r)πℜζ12k2)exp(π(22j-r-1)12ℜ1ζ)≦|ζ|1/22r/2exp(2πn).Thus, for1≦j≦⌊r/2⌋,(3.26)|ℐj,2|≦22kN(2kN)1/22r/2e2πn≦Cj′k3/2N-3/2for a constantCj′  (recalling that n and r are fixed).


## 3.7.2. Estimation ofℐj,1 for 1≦j≦⌊r/2⌋

We have the absolute value of the integrand:(3.27)|ζ2r-jexp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)|×|-1+f(exp(-2π/ζ+2Hjπi/k))f(exp(-22j-r+1π/ζ+22j-r+1Hjπi/k))f(exp(-4πζ+4Hjπik))|=|ζ2r-jexp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)|×|-1+f2j-r(exp(-2πζ+2Hjπik))|=|ζ|1/22(r-j)/2exp((24n+1-2r)πℜζ12k2)exp((22j-r-1)π12ℜ1ζ)×|∑m=1∞p2j-r(m)exp(-2πmζ+2Hjπimk)|≦|ζ|1/22(r-j)/2exp((24n+1-2r)π12)∑m=1∞p2j-r(m)exp(-π12(24m-22j-r+1))≦|ζ|1/22r/2e2πn∑m=1∞p0(m)e-2πm=cj|ζ|1/2for a constantcj. So, for 1≦j≦⌊r/2⌋,(3.28)|ℐj,1|≦22kN(2kN)1/2cj<Cjk3/2N-3/2for a constantCj.


## 3.7.3. Estimation ofI0,1

Letpr*(x) be defined by(3.29)∑n=0∞pr*(n)xn=f(x2r)f(x)f(x2r-1).Again, the regularity of the integrand allows us to alter the path of integration from the arc connectingζI(h,k) and ζT(h,k) to the directed segment.With this in mind, we estimate the absolute value of the integrand:(3.30)|exp(π(1+21-r)24ζ+π(24n+1-2r)ζ12k2)ζ2r-1|×|-1+f(exp(-2π/ζ+2H0πi/k))f(exp(-21-rπ/ζ+21-rH0πi/k))f(exp(-π/ζ+H0πi/k))|=|exp(π(1+21-r)24ζ+π(24n+1-2r)ζ12k2)ζ2r-1|×|∑m=1∞p*(m)exp(-21-rπmζ+21-rH0πimk)|=exp(π(1+21-r)24ℜ1ζ)exp(π(24n+1-2r)ℜζ12k2)|ζ|1/22(r-1)/2×|∑m=1∞p*(m)exp(-21-rπmζ)exp(21-rH0πimk)|≦e2πn|ζ|1/22(r-1)/2×∑m=1∞|p*(m)|exp(π(1+21-r)24ℜ1ζ-21-rπmℜ1ζ)=e2πn|ζ|1/22(r-1)/2∑m=1∞|p*(m)|exp(-π24·2r-1ℜ1ζ(24m-1-2r-1))≦e2πn|ζ|1/22(r-1)/2∑m=1∞|p*(m)|exp(-π24(24m-1-2r-1))<e2πn|ζ|1/22(r-1)/2∑m=1∞|p*(24m-1-2r-1)|y24m-1-2r-1(wherey=e-π/24)=c0|ζ|1/2for a constantc0. So,(3.31)|ℐ0,1|≦22kN(2kN)1/2c0<C0k3/2N-3/2for a constantC0.


## 3.7.4. Estimation ofℐj,1 for 1+⌊r/2⌋≦j≦r≦4

Again, the regularity of the integrand allows us to alter the path of integration from the arc connectingζI(h,k) and ζT(h,k) to the directed segment.With this in mind,(3.32)|exp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)ζ2r-j|×|-1+f2j-r(exp(-2πζ+2Hjπik))|=|exp(π(22j-r-1)12ζ)exp(π(24n+1-2r)ζ12k2)ζ2r-j|×|∑m=1∞p2j-r(m)exp(-2πmζ)exp(2Hjπimk)|=exp(π(22j-r-1)12ℜ1ζ)exp(π(24n+1-2r)ℜζ12k2)|ζ|1/22(r-j)/2×|∑m=1∞p2j-r(m)exp(-2πmℜ1ζ)exp(2Hjπimk)|≦e2πn|ζ|1/22r/2∑m=1∞p2j-r(m)exp(-π12ℜ1ζ(24m-22j-r+1))≦e2πn|ζ|1/22r/2∑m=1∞p0(m)exp(-π12ℜ1ζ(24m-2r+1))=cj|ζ|1/2
for a constant cj. So,(3.33)|ℐj,1|≦22kN(2kN)1/2cj<Cjk3/2N-3/2for a constantCj, when 1+⌊r/2⌋≦j≦r.


## 3.7.5. Combining the Estimates

One has(3.34)|i∑k=1N(k,2r)=1k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)ℐ0,1+i∑j=1⌊r/2⌋∑k=1N(k,2r)=2jk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)(ℐj,1+ℐj,2)+i∑j=1+⌊r/2⌋r∑k=1N(k,2r)=2jk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)ℐj,1|<∑j=0r∑k=1N∑h=0k-1Cjk-1N-3/2+∑j=1⌊r/2⌋∑k=1N∑h=0k-1C'jk-1N-3/2≦C′′N-32∑k=1N1,(whereC′′=∑j=0rCj+∑j=1⌊r/2⌋Cj′)=O(N-1/2).
Thus, we may revise (3.21) to(3.35)pr(n)=i∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)ℐ0,2+i∑j=1+⌊r/2⌋r∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)ℐj,2+O(N-1/2).


## 3.8. Evaluation ofℐj,2 for j=0 and 1+⌊r/2⌋≦j≦r

Writeℐj,2 as(3.36)ℐj,2=∫K(-)exp(π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2)ζ2r-jdζ-ℐj,3-ℐj,4,
where(3.37)ℐj,3:=∫0ζI(h,k),ℐj,4:=∫ζT(h,k)0,
and ℐj,3 and ℐj,4 have the same integrand as (3.36) (and analogously for I0,2).


### 3.8.1. Estimation ofℐj,3 and ℐj,4

We note that the length of the arc of integration inℐj,3 is less than πk/2N, and on this arc|ζ|<2k/N. [50, page 272]. Also, ℜ1/ζ=1 on K [50, page 271, equation (120.2)]. Further, 0<ℜζ<2k2/N2 [50, page 271, equation (119.6)]. The absolute value of the integrand is thus
(3.38)|2r-jζ|1/2exp((24n+1-2r)πℜζ12k2+(22j-r-1)π12ℜ1ζ)<2(r-j)/221/4k1/2N-1/2exp((24n+1-2r)π6N2+(22j-r-1)π12)
so that(3.39)|ℐj,2|<πk2-1/2N-12(r-j)/221/4k1/2N-1/2exp((24n+1-2r)π6N2+(22j-r-1)π12)=πk3/2N-3/22(2r-2j-1)/4exp((24n+1-2r)π6N2+(22j-r-1)π12)=O(k3/2N-3/2exp((24n+1-2r)π6N2)).By the same reasoning,|ℐj,3|=O(k3/2N-3/2exp((24n+1-2r)π/6N2)).We may therefore revise (3.35) to
(3.40)pr(n)=i∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)×∫K(-)ζ2r-1exp{π(21-r+1)24ζ+π(24n+1-2r)ζ12k2}dζ+i∑j=1+⌊r/2⌋r∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)×∫K(-)ζ2r-jexp{π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2}dζ+O(N-1/2)
and upon letting N tend to infinity, obtain(3.41)pr(n)=i∑k=1∞k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)×∫K(-)ζ2r-1exp{π(21-r+1)24ζ+π(24n+1-2r)ζ12k2}dζ+i∑j=1+⌊r/2⌋r∑k=1∞k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)×∫K(-)ζ2r-jexp{π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2}dζ.


## 3.8.1. Estimation ofℐj,3 and ℐj,4

We note that the length of the arc of integration inℐj,3 is less than πk/2N, and on this arc|ζ|<2k/N. [50, page 272]. Also, ℜ1/ζ=1 on K [50, page 271, equation (120.2)]. Further, 0<ℜζ<2k2/N2 [50, page 271, equation (119.6)]. The absolute value of the integrand is thus
(3.38)|2r-jζ|1/2exp((24n+1-2r)πℜζ12k2+(22j-r-1)π12ℜ1ζ)<2(r-j)/221/4k1/2N-1/2exp((24n+1-2r)π6N2+(22j-r-1)π12)
so that(3.39)|ℐj,2|<πk2-1/2N-12(r-j)/221/4k1/2N-1/2exp((24n+1-2r)π6N2+(22j-r-1)π12)=πk3/2N-3/22(2r-2j-1)/4exp((24n+1-2r)π6N2+(22j-r-1)π12)=O(k3/2N-3/2exp((24n+1-2r)π6N2)).By the same reasoning,|ℐj,3|=O(k3/2N-3/2exp((24n+1-2r)π/6N2)).We may therefore revise (3.35) to
(3.40)pr(n)=i∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)×∫K(-)ζ2r-1exp{π(21-r+1)24ζ+π(24n+1-2r)ζ12k2}dζ+i∑j=1+⌊r/2⌋r∑k=1Nk-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)×∫K(-)ζ2r-jexp{π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2}dζ+O(N-1/2)
and upon letting N tend to infinity, obtain(3.41)pr(n)=i∑k=1∞k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)×∫K(-)ζ2r-1exp{π(21-r+1)24ζ+π(24n+1-2r)ζ12k2}dζ+i∑j=1+⌊r/2⌋r∑k=1∞k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)×∫K(-)ζ2r-jexp{π(22j-r-1)12ζ+π(24n+1-2r)ζ12k2}dζ.


## 3.9. The Final Form

We may now introduce the change of variable(3.42)ζ=π(2+22j-r+1-(2-δj0)2)24t
(where the first summation in (3.41) is the j=0 term separated out for clarity), which allows the integral to be evaluated in terms of I3/2, the Bessel function of the first kind of order 3/2 with purely imaginary argument [53, page 372, §17.7] when we bear in mind that a “bent” path of integration is allowable according to the remark preceding Equation (8) on page 177 of [54]. See also [51, page 109]. The final form of the formula is then obtained by using the fact that Bessel functions of half-odd integer order can be expressed in terms of elementary functions.We therefore have, forc=π(2+22j-r+1-(δj0)2)/24,(3.43)pr(n)=i∑k=1∞k-5/2∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)2(r-1)/2(π21-r+124)3/2×∫c-i∞c+i∞t-5/2exp{t+(1+21-r)(24n+1-2r)π2288k2t}dt+i∑j=1+⌊r/2⌋r∑k=1∞k-5/2×∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)×2(r-j)/2(π22j-r-112)3/2×∫c-i∞c+i∞t-5/2exp{t+(22j-r-1)(24n-2r+1)π2144k2t}dt=π(24n-2r+1)3/4×{(1+2r-1)3/42(r-2)/4∑k≧1(k,2max(r,1))=1)k-1×∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2rh,k)ω(2h,k)I3/2(π(24n-2r+1)(1+2r-1)2r/2·6k)+∑j=1+⌊r/2⌋r(22j-r-1)3/42(2-j+r)/2∑k≧1(k,2r)=2jk-1×∑0≦h<k(h,k)=1e-2πinh/kω(h,k)ω(2r-jh,2-jk)ω(h,k/2)I3/2(π(24n-2r+1)(1+22j-r)6k)},
which, after application of the formula [51, page 110]
(3.44)I3/2(z)=2zπddz(sinhzz),
is equivalent to Theorem 2.1.



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*Source: 630458-2010-03-25.xml*