Asymptotics for the Taylor coefficients of certain infinite products

TL;DR

This paper investigates the asymptotic behavior of Taylor coefficients of specific infinite products involving multiple sequences of integers, extending understanding of their growth and distribution.

Contribution

It provides new asymptotic formulas for the Taylor coefficients of complex infinite products with multiple parameters, advancing the analysis of such functions.

Findings

Derived explicit asymptotic formulas for the coefficients

Extended previous results to more general infinite products

Enhanced understanding of the growth rates of these coefficients

Abstract

Let $(m_1,\ldots,m_J)$ and $(r_1,\ldots,r_J)$ be two sequences of $J$ positive integers satisfying $1\le r_j< m_j$ for all $j=1,\ldots,J$. Let $(\delta_1,\ldots,\delta_J)$ be a sequence of $J$ nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product $$\prod_{j=1}^J\Bigg(\prod_{k\ge 1}\big(1-q^{r_j+m_j(k-1)}\big)\big(1-q^{-r_j+m_jk}\big)\Bigg)^{\delta_j}.$$

Equations

$$\prod_{j=1}^{J}\Bigg{(}\prod_{k\geq 1}\big{(}1-q^{r_{j}+m_{j}(k-1)}\big{)}\big{(}1-q^{-r_{j}+m_{j}k}\big{)}\Bigg{)}^{\delta_{j}}.$$

$$(\alpha ;q{)}_n:=\prod _{k=0}^{n-1}(1-\alpha q^k)\text{and}(\alpha ;q{)}_{\infty }:=\prod _{k\ge 0}(1-\alpha q^k).$$

$$(\alpha ,\beta ,\dots ,\gamma ;q{)}_{\infty }:=(\alpha ;q{)}_{\infty }(\beta ;q{)}_{\infty }\cdots (\gamma ;q{)}_{\infty }.$$

$$\sum _{n\ge 0}p(n)q^n=\frac{1}{(q;q{)}_{\infty }}.$$

$$p(n)=\frac{1}{2\sqrt{2}\pi }\sum _{k\ge 1}{A}_k(n)\sqrt{k}\frac{d}{dn}\left(\frac{2}{\sqrt{n-\frac{1}{24}}}\sinh \left(\frac{\pi }{k}\sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right)\right),$$

$$A_k(n)=\sum _{\begin{array}{c}0\le h<k\\ \gcd (h,k)=1\end{array}}{e}^{\pi i(s(h,k)-2nh/k)}$$

$$\frac{1}{(q,q^4;q^5{)}_{\infty }}=\sum _{n\ge 0}\frac{q^{n^2}}{(q;q{)}_n}.$$

$$\displaystyle p_{5,\pm 1}(n)\sim\frac{\csc(\pi/5)}{4\cdot 3^{1/4}\cdot 5^{1/4}}n^{-3/4}\exp\Bigg{(}2\pi\sqrt{\frac{n}{15}}\Bigg{)}.$$

$$R(q):=\frac{q^{1/5}}{1+\frac{q}{1+\frac{q^2}{1+\frac{q^3}{1+\cdots }}}}=q^{1/5}\frac{(q,q^4;q^5{)}_{\infty }}{(q^2,q^3;q^5{)}_{\infty }}.$$

$$\sum _{n\ge 0}C(n)q^n=\frac{(q,q^4;q^5{)}_{\infty }}{(q^2,q^3;q^5{)}_{\infty }}.$$

$$\displaystyle C(n)\sim\frac{2^{1/2}}{5^{3/4}}\cos\Bigg{(}\frac{4\pi}{5}\bigg{(}n+\frac{3}{20}\bigg{)}\Bigg{)}n^{-3/4}\exp\Bigg{(}\frac{4\pi}{5}\sqrt{\frac{n}{5}}\Bigg{)}.$$

$$\prod _{j=1}^{m-1}(q^j;q^m{)}^{-\zeta \chi (j)}$$

$$\sum _{n\ge 0}B(n)q^n=(-q^2,-q^3;q^5{)}_{\infty }^2(q^2,q^8;q^{10}{)}_{\infty }=\frac{(q^2,q^8;q^{10}{)}_{\infty }(q^4;q^6;q^{10}{)}_{\infty }^2}{(q^2,q^3;q^5{)}_{\infty }^2}$$

$$\sum _{n\ge 0}g(n)q^n=\prod _{j=1}^J(q^{r_j},q^{m_j-r_j};q^{m_j}{)}_{\infty }^{{\delta }_j}.$$

$${\lambda }_{m,r}(h,k):=\lceil \frac{rh}{\gcd (m,k)}\rceil$$

$${\lambda }_{m,r}^{\ast }(h,k):={\lambda }_{m,r}(h,k)-\frac{rh}{\gcd (m,k)}.$$

$${\hslash }_m(h,k)\frac{mh}{\gcd (m,k)}\equiv -1(\mathrm{mod}\frac{k}{\gcd (m,k)}).$$

$$\Omega:=\sum_{j=1}^{J}\delta_{j}\Bigg{(}2m_{j}-12r_{j}+\frac{12r_{j}^{2}}{m_{j}}\Bigg{)},$$

$$\Delta(h,k):=-\sum_{j=1}^{J}\delta_{j}\Bigg{(}\frac{2\gcd^{2}(m_{j},k)}{m_{j}}+\frac{12\gcd^{2}(m_{j},k)}{m_{j}}({\lambda^{*}}^{2}_{\!\!\!m_{j},r_{j}}(h,k)-\lambda^{*}_{m_{j},r_{j}}(h,k))\Bigg{)}$$

$${\omega }_{h,k}:=\exp \left(-\pi i\sum _{j=1}^J{\delta }_j\cdot s\left(\frac{m_{j}h}{\gcd (m_j,k)},\frac{k}{\gcd (m_j,k)}\right)\right),$$

$$s(d,c):=\sum_{n\bmod{c}}\bigg{(}\!\!\bigg{(}\frac{dn}{c}\bigg{)}\!\!\bigg{)}\bigg{(}\!\!\bigg{(}\frac{n}{c}\bigg{)}\!\!\bigg{)}$$

$$((x)):=\{\begin{array}{ll}x-\lfloor x\rfloor -1/2& \text{if x∈Z},\\ 0& \text{if x∈Z}.\end{array}$$

$${\text{frenchspacingD}}_{h,k}$$

$$\displaystyle\ \quad+\frac{\hbar_{m_{j}}(h,k)\gcd(m_{j},k)}{k}(\lambda^{2}_{m_{j},r_{j}}(h,k)-\lambda_{m_{j},r_{j}}(h,k))\bigg{)}\Bigg{)}.$$

$$\displaystyle\Pi_{h,k}:=\begin{cases}\displaystyle\prod_{j:\lambda^{*}_{m_{j},r_{j}}(h,k)=0}\Bigg{(}1-\exp\bigg{(}2\pi i\frac{r_{j}\gcd(m_{j},k)+r_{j}\hbar_{m_{j}}(h,k)m_{j}h}{m_{j}k}\bigg{)}\Bigg{)}^{\delta_{j}}\\ \hfill\text{{if there exists $j$ such that $\lambda^{*}_{m_{j},r_{j}}(h,k)=0$}},\\ \quad\quad\ \ 1\\ \hfill\text{otherwise}.\end{cases}$$

$$1-\exp\bigg{(}2\pi i\frac{r_{j}\gcd(m_{j},k)+r_{j}\hbar_{m_{j}}(h,k)m_{j}h}{m_{j}k}\bigg{)}\neq 0.$$

$${\rm Y}(x):=\{\begin{array}{ll}1& \text{if x=0},\\ x& \text{if 0<x≤1/2},\\ 1-x& \text{if 1/2<x<1}.\end{array}$$

$${\mathcal{L}}_{>0}$$

$${\mathcal{L}}_{\le 0}$$

$$\min_{1\leq j\leq J}\left(\Upsilon\big{(}\lambda^{*}_{m_{j},r_{j}}(\varkappa,\ell)\big{)}\frac{\gcd^{2}(m_{j},\ell)}{m_{j}}\right)\geq\frac{\Delta(\varkappa,\ell)}{24}$$

$$g(n)$$

Full text

[TABLE]

Keywords. Infinite product, Taylor coefficient, asymptotics, circle method.

2010MSC. 11P55.

1. Introduction

1.1. Motivations

For complex variables $\alpha$ and $q$ with $|q|<1$, we denote

[TABLE]

We also use the notation

[TABLE]

Let $p(n)$ be the number of partitions of $n$; that is, the number of representations of $n$ written as a sum of a non-increasing sequence of positive integers. It is well known that $p(n)$ has the generating function

[TABLE]

The study of the asymptotic behavior of $p(n)$ originates from Hardy and Ramanujan [9]. A couple of decades later, Rademacher [21] further proved the following formula

[TABLE]

where

[TABLE]

with $s(h,k)$ being the Dedekind sum defined in (1.6).

Apart from ordinary partitions, partitions under symmetric congruence conditions also attract broad research interest. The most famous examples arise from the Rogers–Ramanujan identities (Rogers [25], Ramanujan [23]). Here the first Rogers–Ramanujan identity states that (cf. Corollary 7.67 in [3])

[TABLE]

Using the language in partition theory, the above identity can be restated as follows. The number of partitions of $n$ such that each part is congruent to $\pm 1$ modulo $5$ equals the number of partitions of $n$ such that the adjacent parts differ by at least two. Let $p_{5,\pm 1}(n)$ be the number of partitions of $n$ such that each part is congruent to $\pm 1$ modulo $5$. Its asymptotic formula was shown by Lehner [15]:

[TABLE]

The interested reader may also refer to Niven [18], Livingood [16], Petersson [19, 20], Grosswald [7], Iseki [12, 13, 14], Hagis Jr. [8], Subrahmanyasastri [26] and so forth for the asymptotic behaviors of other partition functions under symmetric congruence conditions.

As we have seen, the generating function of $p_{5,\pm 1}(n)$ is indeed an infinite product under a symmetric congruence condition. Further, similar infinite products are also of number-theoretic interest. One example is the Rogers–Ramanujan continued fraction defined by

[TABLE]

Let us focus on the infinite product part in $R(q)$ and write

[TABLE]

It is known from Richmond and Szekeres [24] that

[TABLE]

Hence for sufficiently large $n$, $C(5n+0,2)>0$ and $C(5n+1,3,4)<0$. We also remark that in [24], Richmond and Szekeres indeed studied the asymptotic behavior of the Taylor coefficients of the general infinite product

[TABLE]

where $m$ is a positive fundamental discriminant, $\chi(j)=(m|j)$ is the Kronecker symbol and $\zeta$ is either $1$ or $-1$.

Recently, there are a number of papers [2, 4, 11, 17, 27] studying vanishing Taylor coefficients of certain infinite products. For instance, Tang [27] showed that the Taylor coefficients of

[TABLE]

satisfy $B(5n+1)=0$ for all $n\geq 0$. At the end of Tang’s paper, he also provided numerical evidence of the inequalities $B(5n+0,2,3)>0$ and $B(5n+4)<0$ for sufficiently large $n$. Similar numerical evidences are also provided for inequalities of Taylor coefficients of other infinite products.

Motivated by these work, it is natural to investigate a broad family of infinite products. Let $\mathbf{m}=(m_{1},\ldots,m_{J})$ and $\mathbf{r}=(r_{1},\ldots,r_{J})$ be two sequences of $J$ positive integers satisfying $1\leq r_{j}<m_{j}$ for all $j=1,\ldots,J$. Let $\boldsymbol{\updelta}=(\delta_{1},\ldots,\delta_{J})$ be a sequence of $J$ nonzero integers. In this paper, we shall study the asymptotics for the Taylor coefficients of the following infinite product

[TABLE]

1.2. Notation and main result

Let $\mathbf{C}$ be the set of complex numbers and $\mathfrak{H}$ be the upper half complex plane. Let $\gcd$ and $\mathrm{lcm}$ be the greatest common divisor function and least common multiple function, respectively. For a positive integer $n$, we accept the convention that $\gcd(0,n)=n$.

We define the big-$O$ notation as usual: $f(x)=O(g(x))$ means that $|f(x)|\leq Cg(x)$ where $C$ is an absolute constant. Furthermore, $f(x)\ll g(x)$ means that $f(x)=O(g(x))$. Throughout this paper, we always assume that the constant $C$ depends on $\mathbf{m}$, $\mathbf{r}$ and $\boldsymbol{\updelta}$ unless otherwise stated.

Below we assume that $0\leq h<k$ are integers such that $\gcd(h,k)=1$. Let us define auxiliary functions

[TABLE]

and

[TABLE]

We also put $\hbar_{m}(h,k)$ an integer such that

[TABLE]

Notice that one may always find such an integer since $\gcd(h,k)=1$.

Next, we define

[TABLE]

[TABLE]

and

[TABLE]

where $s(d,c)$ is the Dedekind sum defined by

[TABLE]

with

[TABLE]

We also define

[TABLE]

One readily verifies that the choice of $\hbar_{m}(h,k)$ does not affect the value of $\textup{\frenchspacing D}_{h,k}$. At last, we define

[TABLE]

Remark 6.1 tells us that the choice of $\hbar_{m}(h,k)$ also does not affect the value of $\Pi_{h,k}$. Also, Proposition 6.3 indicates that for any $j$ with $\lambda^{*}_{m_{j},r_{j}}(h,k)=0$, we have

[TABLE]

Hence the value $\Pi_{h,k}$ is well-defined and $\Pi_{h,k}\neq 0$.

Given a real $0\leq x<1$, we define

[TABLE]

Let $L=\mathrm{lcm}(m_{1},\ldots,m_{R})$. We define two disjoint sets:

[TABLE]

Our main result states as follows.

Theorem 1.1**.**

If the inequality

[TABLE]

holds for all $1\leq\ell\leq L$ and $0\leq\varkappa<\ell$, then for positive integers $n>-\Omega/24$, we have

[TABLE]

where

[TABLE]

$I_{s}(x)$* is the modified Bessel function of the first kind, and*

[TABLE]

Remark 1.1*.*

To better understand the asymptotic behavior of $g(n)$, one may apply the asymptotic expansion of $I_{s}(x)$ (cf. [1, p. 377, (9.7.1)]): for fixed $s$, when $|\arg x|<\frac{\pi}{2}$,

[TABLE]

2. Applications of the main result

Before moving to the proof of the main result, we first give some applications. In the first two examples, we reproduce the asymptotic formulas (1.2) and (1.3), respectively. We then confirm Tang’s inequalities in [27] in the asymptotic sense. In this section, we always expand the infinite product as $\sum_{n\geq 0}g(n)q^{n}$.

In general, to obtain an explicit asymptotic formula of $g(n)$, we first compute $\mathcal{L}_{>0}$. Next, we find the largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ with $(\varkappa,\ell)\in\mathcal{L}_{>0}$ and $k\equiv\ell\pmod{L}$. Now one needs to check if the corresponding $I$-Bessel function vanishes for this choice. If it is nonvanishing, then the asymptotic formula shall be obtained from the $I$-Bessel term. Otherwise, we move to find the second largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ and carry out the same program. Notice that if there are multiple choices of $\varkappa$, $\ell$ and $k$ giving the same value of $\sqrt{\Delta(\varkappa,\ell)}/k$, one should sum up all such $I$-Bessel terms and check if the summation vanishes or not.

2.1. Partitions into parts congruent to $\pm 1$ modulo $5$

Let

[TABLE]

Then $\mathbf{m}=\{5\}$, $\mathbf{r}=\{1\}$ and $\boldsymbol{\updelta}=\{-1\}$. Hence $L=5$ and $\Omega=-2/5$. We now compute that

[TABLE]

First, the assumption (1.7) is satisfied. We next find that the largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ with $(\varkappa,\ell)\in\mathcal{L}_{>0}$ and $k\equiv\ell\pmod{L}$ is $\sqrt{\frac{2}{5}}$. Here we have two choices:

[TABLE]

When $k=1$, the admissible $(h,k)$ is $(0,1)$. We compute that the $I$-Bessel term is

[TABLE]

When $k=5$, noticing that $\gcd(0,5)=5\neq 1$, there is no admissible $(h,k)$. Hence,

[TABLE]

2.2. The Rogers–Ramanujan continued fraction

Let

[TABLE]

Then $\mathbf{m}=\{5,5\}$, $\mathbf{r}=\{1,2\}$ and $\boldsymbol{\updelta}=\{1,-1\}$. Hence $L=5$ and $\Omega=24/5$. We compute that

[TABLE]

First, the assumption (1.7) is satisfied. We next find that the largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ with $(\varkappa,\ell)\in\mathcal{L}_{>0}$ and $k\equiv\ell\pmod{L}$ is $\frac{2\sqrt{6}}{5\sqrt{5}}$. Here we have two choices:

[TABLE]

When $k=5$, the admissible $(h,k)$ are $(2,5)$ and $(3,5)$. We compute that, in total, the $I$-Bessel term is

[TABLE]

Notice that \cos\big{(}\frac{4\pi}{5}(n+\frac{3}{20})\big{)} does not vanish for all $n$. Hence,

[TABLE]

2.3. Tang’s inequalities

Let

[TABLE]

Then $\mathbf{m}=\{5,10,10\}$, $\mathbf{r}=\{2,2,4\}$ and $\boldsymbol{\updelta}=\{-2,1,2\}$. Hence $L=10$ and $\Omega=-8$. We compute that

[TABLE]

First, the assumption (1.7) is satisfied. We next find that the largest number among $\{\sqrt{\Delta(\varkappa,\ell)}/k\}$ with $(\varkappa,\ell)\in\mathcal{L}_{>0}$ and $k\equiv\ell\pmod{L}$ is $\frac{1}{\sqrt{5}}$. Here we have four choices:

[TABLE]

When $k=1$, the admissible $(h,k)$ is $(0,1)$. We compute that the $I$-Bessel term is

[TABLE]

When $k=5$, the admissible $(h,k)$ are $(2,5)$ and $(3,5)$. We compute that, in total, the $I$-Bessel term is

[TABLE]

In total, we therefore have

[TABLE]

Notice that \sin\big{(}\frac{\pi}{5}\big{)}+\sin\big{(}\frac{2\pi}{5}(2n+1)\big{)} vanishes only if $n\equiv 1\pmod{5}$. Hence, for $n\not\equiv 1\pmod{5}$,

[TABLE]

It follows that $g(5n+0,2,3)>0$ and $g(5n+4)<0$ for sufficiently large $n$. If we further compute a number of lower $I$-Bessel terms, we still encounter the same vanishment for $n\equiv 1\pmod{5}$. This highly suggests that $g(5n+1)=0$, which is, indeed, proved by Tang using elementary techniques in [27].

All other inequalities conjectured by Tang can be proved for sufficiently large $n$ in the same manner. We omit the details here.

3. Dedekind eta function and Jacobi theta function

In this section, we introduce the Dedekind eta function and Jacobi theta function. All results here are standard, which can be found in, for example, [5] or [28].

Let $\tau\in\mathfrak{H}$ and $\varsigma\in\mathbf{C}$. The Dedekind eta function is defined by

[TABLE]

with $q:=e^{2\pi i\tau}$. Further, the Jacobi theta function reads

[TABLE]

Notice that if we put $\zeta:=e^{2\pi i\varsigma}$, then the Jacobi triple product identity indicates that

[TABLE]

It follows immediately that

[TABLE]

The Dedekind eta function and Jacobi theta function are of broad interest due to their transformation properties. Let $\displaystyle\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL_{2}(\mathbb{Z})$ where we assume that $c>0$. Recall that the Möbius transformation for $\tau\in\mathfrak{H}$ is defined by

[TABLE]

Further, for the $\gamma$ given above, we write for convenience

[TABLE]

If

[TABLE]

where, again, $s(d,c)$ is the Dedekind sum, then

[TABLE]

and

[TABLE]

Further, let $\alpha$ and $\beta$ be integers. The Jacobi theta function also satisfies

[TABLE]

4. Farey arcs and a transformation formula

To study the asymptotics for the Taylor coefficients of $G(q)$, we turn to the celebrated circle method due to Rademacher [21, 22] whose idea originates from Hardy and Ramanujan [9]. Recalling that $G(q)$ is holomorphic inside the unit disk, we may directly apply Cauchy’s integral formula to deduce

[TABLE]

where the contour integral is taken counter-clockwise. Now one puts $r=e^{-2\pi\varrho}$ with $\varrho=1/N^{2}$ where $N$ is a sufficiently large positive integer.

Next, we dissect the circle $\mathcal{C}$ by Farey arcs. Let $h/k$ with $\gcd(h,k)=1$ be a Farey fraction of order $N$. If we denote by $\xi_{h,k}$ the interval $[-\theta^{\prime}_{h,k},\theta^{\prime\prime}_{h,k}]$ with $\theta^{\prime}_{h,k}$ and $\theta^{\prime\prime}_{h,k}$ being the positive distances from $h/k$ to its neighboring mediants, then

[TABLE]

Making the changes of variables $z=k(\varrho-i\phi)$ and $\tau=(h+iz)/k$ yields

[TABLE]

Let $r<m$ be positive integers. Our next task is to apply the transformation properties of the Dedekind eta function and Jacobi theta function so that $\textup{\frenchspacing Zh}(r\tau;m\tau)$ can be nicely reformulated around the Farey arc with respect to $h/k$. To do so, we need to construct a suitable matrix in $SL_{2}(\mathbb{Z})$.

Let $d=\gcd(m,k)$. For convenience, we write $m=dm^{\prime}$ and $k=dk^{\prime}$. Recalling that $\hbar_{m}(h,k)$ satisfies $\hbar_{m}(h,k)m^{\prime}h\equiv-1\pmod{k^{\prime}}$, we put $b_{m^{\prime}}=(\hbar_{m}(h,k)m^{\prime}h+1)/k^{\prime}$. It is straightforward to verify that the following matrix is in $SL_{2}(\mathbb{Z})$:

[TABLE]

Since $\tau=(h+iz)/k=(h+iz)/dk^{\prime}$, one may compute

[TABLE]

Namely,

[TABLE]

On the other hand, we have

[TABLE]

and hence

[TABLE]

Further,

[TABLE]

Recalling from (3.1) that

[TABLE]

one has, from (3.2), (3.3), (3.4) and the fact $s(-m^{\prime}h,k^{\prime})=-s(m^{\prime}h,k^{\prime})$, that

[TABLE]

Consequently, we deduce the following transformation formula.

[TABLE]

Remark 4.1*.*

It follows from (4.5) that for all $j=1,2,\ldots,J$,

[TABLE]

5. Some auxiliary results

5.1. Necessary bounds

Now we are going to present some useful bounds, which were obtained in the previous work; see, for example, [21].

First, it is well known (cf. Chapter 3 in [10]) that for a Farey fraction $h/k$ of order $N$, one has

[TABLE]

and hence

[TABLE]

Next, since $z=k(\varrho-i\phi)$, it follows that

[TABLE]

Further, one has

[TABLE]

since

[TABLE]

where we use the fact $k\leq N$ in the last inequality.

5.2. A partition-theoretic result

Let $\eta$ be a positive integer. Let $p^{*}_{\eta}(s,t;n)$ denote the number of 2-colored (say, red and blue) partition $\eta$-tuples of $n$ with $s$ parts in total colored by red and $t$ parts in total colored by blue. Here we allow [math] as a part. Let $q$, $\zeta$ and $\xi$ be such that $|q|<1$, $|\zeta|<1$ and $|\xi|<1$. The following infinite triple summation

[TABLE]

is absolutely convergent. Further, considering another absolutely convergent infinite triple summation

[TABLE]

an easy partition-theoretic argument indicates that $|d^{*}_{\eta}(s,t;n)|\leq p^{*}_{\eta}(s,t;n)$ for all $s,t,n\geq 0$. Also, we have $d^{*}_{\eta}(0,0;0)=p^{*}_{\eta}(0,0;0)=1$.

In general, for a nonzero integer $\delta$, if we write

[TABLE]

then

[TABLE]

and hence $|a_{\delta}(s,t;n)|\leq p^{*}_{|\delta|}(s,t;n)$ for all $s,t,n\geq 0$. Trivially, we also have

[TABLE]

Further, for real $0\leq\alpha,\beta,x<1$, we have

[TABLE]

6. Outline of the proof

We know from (4.1) and (4.6) that

[TABLE]

Let us fix a Farey fraction $h/k$. We first find integers $1\leq\ell\leq L$ and $0\leq\varkappa<\ell$ such that $k\equiv\ell\pmod{L}$ and $h\equiv\varkappa\pmod{\ell}$. For convenience, we write $\rho(h,k):=(\varkappa,\ell)$. It is not hard to observe that for all $j=1,2,\ldots,J$,

[TABLE]

It turns out that $\Delta(h,k)=\Delta(\varkappa,\ell)$. We now split $g(n)$ as follows.

[TABLE]

The minor arcs are those with respect to $h/k$ with $\rho(h,k)\in\mathcal{L}_{\leq 0}$. We have the following bound.

Theorem 6.1**.**

Let $(\varkappa,\ell)\in\mathcal{L}_{\leq 0}$. For positive integers $n>-\Omega/24$, we have

[TABLE]

In particular, if we take $N=\left\lfloor\sqrt{2\pi\left(n+\frac{\Omega}{24}\right)}\right\rfloor$, then $S_{\varkappa,\ell}\ll_{\mathbf{m},\mathbf{r},\boldsymbol{\updelta}}1$.

The arcs with respect to $h/k$ with $\rho(h,k)\in\mathcal{L}_{>0}$ give us the main contribution.

Theorem 6.2**.**

Let $(\varkappa,\ell)\in\mathcal{L}_{>0}$. If the inequality

[TABLE]

holds, then for positive integers $n>-\Omega/24$, we have

[TABLE]

where

[TABLE]

In particular, if we take $N=\left\lfloor\sqrt{2\pi\left(n+\frac{\Omega}{24}\right)}\right\rfloor$, then $E_{\varkappa,\ell}\ll_{\mathbf{m},\mathbf{r},\boldsymbol{\updelta}}1$.

Theorems 6.1 and 6.2 immediately imply the main result. Before presenting proofs of the two results respectively in Sections 7 and 8, we make the following preparations.

For fixed $\varkappa$ and $\ell$ with $1\leq\ell\leq L$ and $0\leq\varkappa<\ell$, one may split the indexes $\{1,2,\ldots,J\}$ into two disjoint parts:

[TABLE]

so that for $j^{*}\in\mathcal{J}^{*}_{\varkappa,\ell}$ we have $\lambda^{*}_{m_{j^{*}},r_{j^{*}}}(\varkappa,\ell)=0$ and for $j^{**}\in\mathcal{J}^{**}_{\varkappa,\ell}$ we have $\lambda^{*}_{m_{j^{**}},r_{j^{**}}}(\varkappa,\ell)\neq 0$.

Proposition 6.3**.**

Let $j^{*}\in\mathcal{J}^{*}_{\varkappa,\ell}$. For any Farey fraction $h/k$ such that $k\equiv\ell\pmod{L}$ and $h\equiv\varkappa\pmod{\ell}$, we have that

[TABLE]

is a real noninteger. Further,

[TABLE]

Proof.

In this proof, we write for short $m=m_{j^{*}}$ and $r=r_{j^{*}}$. We also write $d=\gcd(m,k)$, $m=dm^{\prime}$ and $k=dk^{\prime}$. Since $j^{*}\in\mathcal{J}^{*}_{\varkappa,\ell}$, we have $\lambda^{*}_{m,r}(h,k)=\lambda^{*}_{m,r}(\varkappa,\ell)=0$. Hence $d$ divides $rh$ and $\lambda_{m,r}(h,k)=rh/d$. We know from (4.5) that

[TABLE]

where as in Section 4, we have put $b_{m^{\prime}}=(\hbar_{m}(h,k)m^{\prime}h+1)/k^{\prime}$. Hence it is a real number.

Notice that $d=\gcd(m,k)$. Since $\gcd(h,k)=1$, $d\mid rh$ implies that $d\mid r$. Further, $b_{m^{\prime}}=(\hbar_{m}(h,k)m^{\prime}h+1)/k^{\prime}$ implies that $\gcd(m^{\prime},b_{m^{\prime}})=1$. Hence, if $\frac{b_{m^{\prime}}}{m^{\prime}}\frac{r}{d}$ is an integer, then $m^{\prime}\mid\frac{r}{d}$ so that $m=dm^{\prime}\mid r$. This violates the assumption that $1\leq r\leq m-1$. Hence $r\tau\gamma^{*}_{(m,h,k)}(m\tau)+\lambda_{m,r}(h,k)\gamma_{(m,h,k)}(m\tau)$ is not an integer and (6.3) follows immediately. ∎

Remark 6.1*.*

Recall that $\hbar_{m}(h,k)$ is defined to be an integer such that

[TABLE]

Let $n$ be an integer. It turns out that

[TABLE]

since from the above proof we have $\gcd(m_{j^{*}},k)\mid r_{j^{*}}$. Hence the choice of $\hbar_{m_{j^{*}}}(h,k)$ does not affect the value of

[TABLE]

7. Minor arcs

Let $(\varkappa,\ell)\in\mathcal{L}_{\leq 0}$, namely, $\Delta(\varkappa,\ell)\leq 0$. We write $\mathcal{J}^{*}=\mathcal{J}^{*}_{\varkappa,\ell}$ and $\mathcal{J}^{**}=\mathcal{J}^{**}_{\varkappa,\ell}$. Notice that

[TABLE]

We now consider the Farey arcs with respect to $h/k$ with $k\equiv\ell\pmod{L}$ and $h\equiv\varkappa\pmod{\ell}$. Since $\Delta(\varkappa,\ell)\leq 0$, it follows from (5.3) and (5.4) that

[TABLE]

For convenience, now we write $\lambda_{j}=\lambda_{m_{j},r_{j}}(h,k)$ and $\lambda^{*}_{j}=\lambda^{*}_{m_{j},r_{j}}(h,k)$. We also write for short $\tilde{\varsigma}_{j}=r_{j}\tau\gamma^{*}_{(m_{j},h,k)}(m_{j}\tau)+\lambda_{m_{j},r_{j}}(h,k)\gamma_{(m_{j},h,k)}(m_{j}\tau)$ and $\tilde{\tau}_{j}=\gamma_{(m_{j},h,k)}(m_{j}\tau)$. We know from (4.3) and (4.5) that

[TABLE]

and

[TABLE]

Notice that

[TABLE]

We write

[TABLE]

First, it follows from Proposition 6.3 that

[TABLE]

Further, as we have seen in Section 5.2, for $j^{*}\in\mathcal{J}^{*}$ (hence $\lambda^{*}_{j^{*}}=0$),

[TABLE]

where we use $\Re(z^{-1})\geq k/2$. It follows from (5.5) that

[TABLE]

Likewise, for $j^{**}\in\mathcal{J}^{**}$,

[TABLE]

Hence,

[TABLE]

8. Major arcs

Let $(\varkappa,\ell)\in\mathcal{L}_{>0}$, namely, $\Delta(\varkappa,\ell)>0$. Again, we write $\mathcal{J}^{*}=\mathcal{J}^{*}_{\varkappa,\ell}$ and $\mathcal{J}^{**}=\mathcal{J}^{**}_{\varkappa,\ell}$. Let us consider the Farey arcs with respect to $h/k$ with $k\equiv\ell\pmod{L}$ and $h\equiv\varkappa\pmod{\ell}$. For convenience, we write $\tilde{\varsigma}_{j}(h,k)=r_{j}\tau\gamma^{*}_{(m_{j},h,k)}(m_{j}\tau)+\lambda_{m_{j},r_{j}}(h,k)\gamma_{(m_{j},h,k)}(m_{j}\tau)$ and $\tilde{\tau}_{j}(h,k)=\gamma_{(m_{j},h,k)}(m_{j}\tau)$.

Recall that

[TABLE]

We split $S_{\varkappa,\ell}$ into two parts $\Sigma_{1}$ and $\Sigma_{2}$ where

[TABLE]

and

[TABLE]

We first show that $\Sigma_{2}$ is negligible. Notice that by (5.3)

[TABLE]

Let us fix $h$ and $k$ and write $\tilde{\varsigma}_{j}=\tilde{\varsigma}_{j}(h,k)$ and $\tilde{\tau}_{j}=\tilde{\tau}_{j}(h,k)$. We also write $\lambda^{*}_{j}=\lambda^{*}_{m_{j},r_{j}}(h,k)$. Recalling the definition of $\Pi_{h,k}$ and Proposition 6.3, we have

[TABLE]

Let us write for short

[TABLE]

It follows again from (4.3) and (4.5) that

[TABLE]

[TABLE]

and

[TABLE]

where for real $0\leq x<1$,

[TABLE]

We have

[TABLE]

Hence,

[TABLE]

Since at least one coordinate of $\mathbf{n}\times\mathbf{s}\times\mathbf{t}$ is nonzero, under the condition (6.1), we know that

[TABLE]

for all $\mathbf{n}\times\mathbf{s}\times\mathbf{t}\in(\mathbb{Z}_{\geq 0}^{J})^{3}\backslash(0,\ldots,0)^{3}$. Recalling that $\Re(z^{-1})\geq k/2$, it follows that

[TABLE]

is maximized when $\Re(z^{-1})=k/2$. Namely,

[TABLE]

Together with the fact $\prod_{h,k}\ll 1$ which follows from (6.3), we conclude that

[TABLE]

Finally, we estimate the main contribution $\Sigma_{1}$. To do so, we need the following evaluation of an integral, which is a special case of Lemma 2.4 in [6]. For the sake of completeness, we sketch a brief proof.

Lemma 8.1**.**

Let $a\in\mathbb{R}_{>0}$ and $b\in\mathbb{R}$. Let $\gcd(h,k)=1$. Define

[TABLE]

Then for positive integers $n$ with $n>-b/24$, we have

[TABLE]

where

[TABLE]

Proof.

Putting $w=z/k=\varrho-i\phi$ and reversing the integral order, one has

[TABLE]

We now split the integral into three parts:

[TABLE]

where

[TABLE]

is a Hankel contour.

The dominant contribution to $I$ comes from $J_{1}$. We make the following change of variables $t=wk\sqrt{(24n+b)/a}$. Then

[TABLE]

in which the new contour $\tilde{\Gamma}$ is still a Hankel contour. Recalling the contour integral representation of $I_{s}(x)$:

[TABLE]

we conclude that

[TABLE]

We next bound the error term $E(I)$, coming from $J_{2}$ and $J_{3}$. Let us put $w=x+i\theta$ with $-\infty\leq x\leq\varrho$ and $\theta\in\{\theta^{\prime}_{h,k},-\theta^{\prime\prime}_{h,k}\}$. We know that

[TABLE]

and

[TABLE]

where we use the fact $\frac{1}{2kN}\leq|\theta|\leq\frac{1}{kN}$. Hence for $j=2$ and $3$, we have

[TABLE]

This implies that

[TABLE]

which gives (8.3). ∎

Recall that

[TABLE]

The main contribution to $\Sigma_{1}$ is

[TABLE]

The error term in $\Sigma_{1}$ is bounded by

[TABLE]

Acknowledgements

I want to thank George Andrews for helpful suggestions and for pointing out a number of useful references.