On the Fourth Power Moment of Fourier Coefficients of Cusp Form

TL;DR

This paper derives an asymptotic formula for the fourth power moment of the partial sums of Fourier coefficients of holomorphic cusp forms, improving previous bounds and deepening understanding of their distribution.

Contribution

It establishes a refined asymptotic formula for the fourth power moment of Fourier coefficient sums, enhancing previous results with a better error term.

Findings

Derived an asymptotic formula for the fourth power moment of A(x)

Improved the error term in the asymptotic estimate

Enhanced understanding of the distribution of Fourier coefficients

Abstract

Let $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight $\kappa=2n\geqslant12$ for the full modular group and $A(x)=\sum\limits_{n\leqslant x}a(n)$. In this paper, we establish an asymptotic formula of the fourth power moment of $A(x)$ and prove that \begin{equation*} \int_1^TA^4(x)\mathrm{d}x=\frac{3}{64\kappa\pi^4}s_{4;2}(\tilde{a}) T^{2\kappa}+O\big(T^{2\kappa-\delta_4+\varepsilon}\big) \end{equation*} with $\delta_4=1/8$, which improves the previous result.

Equations

$$\int_{1}^{T}A^{4}(x)\mathrm{d}x=\frac{3}{64\kappa\pi^{4}}s_{4;2}(\tilde{a})T^{2\kappa}+O\big{(}T^{2\kappa-\delta_{4}+\varepsilon}\big{)}$$

$$a(n)\ll n^{(\kappa -1)/2}d(n),$$

$$A(x):=\sum _{n⩽x}a(n).$$

$$A(x)=\Omega_{\pm}\big{(}x^{\kappa/2-1/4}\log\log\log x\big{)}.$$

$$A(t_1)>B{t}_1^{\kappa /2-1/4},A(t_2)<-B{t}_2^{\kappa /2-1/4},$$

$$A(x)\ll x^{(\kappa -1)/2+1/4+\epsilon }$$

$${\int }_1^T{A}^2(x)\mathrm{d}x={\mathcal{C}}_2{T}^{\kappa +1/2}+B(T),$$

$${\mathcal{C}}_2=\frac{1}{(4\kappa +2){\pi }^2}\sum _{n=1}^{\infty }{a}^2(n)n^{-\kappa -1/2},$$

$$B(T)\ll T^{\kappa}\log^{5}T,\quad B(T)=\Omega\bigg{(}T^{\kappa-1/4}\frac{(\log\log\log T)^{3}}{\log T}\bigg{)}.$$

$${\int }_1^T{A}^8(x)\mathrm{d}x\ll T^{4\kappa -1+\epsilon }.$$

$${\int }_1^T{A}^3(x)\mathrm{d}x={\mathcal{C}}_3{T}^{(6\kappa +1)/4}+O(T^{(6\kappa +1)/4-{\delta }_3+\epsilon }),$$

$${\int }_1^T{A}^4(x)\mathrm{d}x={\mathcal{C}}_4{T}^{2\kappa }+O(T^{2\kappa -{\delta }_4+\epsilon }),$$

$${\mathcal{C}}_3:=\frac{3}{4(6\kappa +1){\pi }^3}\sum _{\begin{array}{c}n,m,k\in \mathbb{N}\\ \sqrt{n}+\sqrt{m}=\sqrt{k}\end{array}}(nmk{)}^{-\kappa /2-1/4}a(n)a(m)a(k),$$

$${\mathcal{C}}_4:=\frac{3}{64\kappa {\pi }^4}\sum _{\begin{array}{c}n,m,k,\ell \in \mathbb{N}\\ \sqrt{n}+\sqrt{m}=\sqrt{k}+\sqrt{\ell }\end{array}}(nmk\ell {)}^{-\kappa /2-1/4}a(n)a(m)a(k)a(\ell ).$$

$$\int_{1}^{T}A^{k}(x)\mathrm{d}x=\mathcal{C}_{k}T^{1+k(2\kappa-1)/4}+O\big{(}T^{1+k(2\kappa-1)/4-\delta_{k}+\varepsilon}\big{)}$$

$$\int_{1}^{T}A^{4}(x)\mathrm{d}x=\frac{3}{64\kappa\pi^{4}}s_{4;2}(\tilde{a})T^{2\kappa}+O\big{(}T^{2\kappa-\delta_{4}+\varepsilon}\big{)}$$

$$s_{4;2}(\tilde{a})=\sum _{\begin{array}{c}n,m,k,\ell \in {\mathbb{N}}^{\ast }\\ \sqrt{n}+\sqrt{m}=\sqrt{k}+\sqrt{\ell }\end{array}}\frac{a(n)a(m)a(k)a(\ell )}{(nmk\ell {)}^{\kappa /2+1/4}}.$$

$$s_{k;\ell }(f):=\sum _{\begin{array}{c}{n}_1,\cdots ,n_{\ell },n_{\ell +1},\cdots ,n_k\in {\mathbb{N}}^{\ast }\\ \sqrt{n_1}+\cdots +\sqrt{n_l}=\sqrt{n_{\ell +1}}+\cdots +\sqrt{n_k}\end{array}}\frac{f(n_1)f(n_2)\cdots f(n_k)}{(n_1{n}_2\cdots n_k{)}^{3/4}},1⩽\ell <k.$$

$$s_{k;\ell }(f;y):=\sum _{\begin{array}{c}{n}_1,\cdots ,n_{\ell },n_{\ell +1},\cdots ,n_k⩽y\\ \sqrt{n_1}+\cdots +\sqrt{n_l}=\sqrt{n_{\ell +1}}+\cdots +\sqrt{n_k}\end{array}}\frac{f(n_1)f(n_2)\cdots f(n_k)}{(n_1{n}_2\cdots n_k{)}^{3/4}},1⩽\ell <k.$$

$$\int_{a}^{b}g(x)h(x)\mathrm{d}x\ll\Big{(}\max_{a\leqslant x\leqslant b}|g(x)|\Big{)}\bigg{(}\max_{a\leqslant u<v\leqslant b}\bigg{|}\int_{u}^{v}h(x)\mathrm{d}x\bigg{|}\bigg{)}.$$

$${\int }_T^{2T}{t}^{\alpha }\cos (A\sqrt{t}+B)\mathrm{d}t\ll T^{1/2+\alpha }|A{|}^{-1}.$$

$$|\sqrt{n}+\sqrt{m}\pm \sqrt{k}-\sqrt{\ell }|\gg (nmk\ell {)}^{-1/2}\max (n,m,k,\ell {)}^{-3/2},$$

$$\big{|}s_{k;\ell}(f)-s_{k;\ell}(f;y)\big{|}\ll y^{-1/2+\varepsilon},\quad 1\leqslant\ell<k,$$

$$0<|\sqrt{n}+\sqrt{m}-\sqrt{k}-\sqrt{\ell }|<\Delta$$

$${\mathcal{A}}_1(N,M,K,L;\Delta )\ll \Delta L^{1/2}NMK+NK{L}^{1/2+\epsilon }.$$

$${\mathcal{A}}_1(N,M,K,L;\Delta )\ll \Delta L^{1/2}NMK.$$

$$0<|\sqrt{n_1}+\sqrt{n_2}\pm \sqrt{n_3}-\sqrt{n_4}|<\Delta$$

$$\mathscr{A}_{\pm}(N_{1},N_{2},N_{3},N_{4};\Delta)\ll\prod_{j=1}^{4}\big{(}\Delta^{1/4}N_{j}^{7/8}+N_{j}^{1/2}\big{)}N_{j}^{\varepsilon}.$$

$$A(x)=\frac{1}{\sqrt{2}\pi }\sum _{n⩽N}\frac{a(n)}{n^{\kappa /2+1/4}}{x}^{\kappa /2-1/4}\cos (4\pi \sqrt{nx}-\pi /4)+O(x^{\kappa /2+\epsilon }{N}^{-1/2}),$$

$$A(x)=\frac{1}{\sqrt{2}\pi }\mathcal{R}(x)+O(x^{\kappa /2+\epsilon }{y}^{-1/2}),$$

Full text

On the Fourth Power Moment of Fourier

Coefficients of Cusp Form

Jinjiang Li11footnotemark: 1       &    Panwang Wang 22footnotemark: 2       &    Min Zhang333Corresponding author.

    * E-mail addresses*: jinjiang.li.math@gmail.com (J. Li), panwangw@gmail.com (P. Wang),

          min.zhang.math@gmail.com (M. Zhang).

$\textrm{\small Department of Mathematics, China University of Mining and Technology}^{*\,{\dagger}\,{\ddagger}}$

Beijing 100083, P. R. China

Abstract: Let $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight $\kappa=2n\geqslant 12$ for the full modular group and $A(x)=\sum\limits_{n\leqslant x}a(n)$. In this paper, we establish an asymptotic formula of the fourth power moment of $A(x)$ and prove that

[TABLE]

with $\delta_{4}=1/8$, which improves the previous result.

Keywords: Cusp form; Fourier coefficient; mean value; asymptotic formula

Mathematics Subject Classification 2010: 11N37, 11M06

1 Introduction and main result

Let $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight $\kappa=2n\geqslant 12$ for the full modular group. In 1974, Deligne[2] proved the following profound result

[TABLE]

where $d(n)$ denotes the Dirichlet divisor function and the implied constant in $\ll$ is absolute. Suppose $x\geqslant 2$ and define

[TABLE]

It is well known that $A(x)$ has no main term and $A(x)\ll x^{\kappa/2-1/6+\varepsilon}$. In 1973, Joris[5] proved that

[TABLE]

In 1990, Ivić[3] showed that there exist two points $t_{1}$ and $t_{2}$ in the interval $[T,T+CT^{1/2}]$ such that

[TABLE]

where $B>0,\,C>0$ are constants. It is conjectured that

[TABLE]

is true for every $\varepsilon.$ The evidence in support of this conjecture has been given by Ivić[3], who proved the following square mean value formula of $A(x)$, i.e.

[TABLE]

where

[TABLE]

[TABLE]

In[3], Ivić also proved the upper bound of eighth power moment of $A(x)$, that is

[TABLE]

Cai[1] studied the third and fourth power moments of $A(x)$. He proved that

[TABLE]

[TABLE]

where $\delta_{3}=1/14,\,\delta_{4}=1/23$ and

[TABLE]

[TABLE]

In[10], Zhai proved that (1.3) holds for $\delta_{3}=1/4.$ Following the approach of Tsang[9], Zhai[10] proved that the equation (1.4) holds for $\delta_{4}=2/41.$ This approach used the method of exponential sums. In particular, if the exponent pair conjecture is true, namely, if $(\varepsilon,1/2+\varepsilon)$  is an exponent pair, then the equation (1.4) holds for $\delta_{4}=1/14$. Later, combining the method of [4] and a deep result of Robert and Sargos[8], Zhai[12] proved that the equation (1.4) holds for $\delta_{4}=3/28$. By a unified approach, Zhai[11] proved that the asymptotic formula

[TABLE]

holds for $3\leqslant k\leqslant 7,$ where $\mathcal{C}_{k}$ and $0<\delta_{k}<1$ are explicit constants.

The aim of this paper is to improve the value of $\delta_{4}=3/28$, which is achieved by Zhai[12]. The main result is the following

Theorem 1.1

We have

[TABLE]

with $\delta_{4}=1/8,$ where

[TABLE]

Notation. Throughout this paper, $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight $\kappa=2n\geqslant 12$ for the full modular group; $d(n)$ denote the Dirichlet divisor function; $\tilde{a}(n):=a(n)n^{-\kappa/2+1/2}$; $\|x\|$ denotes the distance from $x$ to the nearest integer, i.e., $\|x\|=\min\limits_{n\in\mathbb{Z}}|x-n|$. $[x]$ denotes the integer part of $x$; $n\sim N$ means $N<n\leqslant 2N$; $n\asymp N$ means $C_{1}N\leqslant n\leqslant C_{2}N$ with positive constants $C_{1},\,C_{2}$ satisfying $C_{1}<C_{2}.$ $\varepsilon$ always denotes an arbitrary small positive constant which may not be the same at different occurances. We shall use the estimates $d(n)\ll n^{\varepsilon}$. Suppose $f:\mathbb{N}\to\mathbb{R}$ is any function satisfying $f(n)\ll n^{\varepsilon},\,k\geqslant 2$ is a fixed integer. Define

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We shall use $s_{k;\ell}(f)$ to denote both of the series (1.5) and its value. Suppose $y>1$ is a large parameter, and we define

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2 Preliminary Lemmas

Lemma 2.1

If $g(x)$ and $h(x)$ are continuous real-valued functions of $x$ and $g(x)$ is monotonic, then

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Proof. See Tsang [9], Lemma 1.

Lemma 2.2

Suppose $A,\,B\in\mathbb{R},\,A\not=0.$ . Then we have

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Proof. It follows from Lemma 2.1 easily.

Lemma 2.3

If $n,m,k,\ell\in\mathbb{N}$ such that $\sqrt{n}+\sqrt{m}\pm\sqrt{k}-\sqrt{\ell}\not=0$, then there hold

[TABLE]

respectively.

Proof. See Kong [7], Lemma 3.2.1.

Lemma 2.4

Let $f:\mathbb{N}\to\mathbb{R}$ be any function satisfying $f(n)\ll n^{\varepsilon}$. Then we have

[TABLE]

where $k\geqslant 2$ is a fixed integer.

Proof. See Zhai [11], Lemma 3.1.

Lemma 2.5

Suppose $1\leqslant N\leqslant M,\,1\leqslant K\leqslant L,\,N\leqslant K,\,M\asymp L,\,0<\Delta\ll L^{1/2}$. Let $\mathscr{A}_{1}(N,M,K,L;\Delta)$ denote the number of solutions of the following inequality

[TABLE]

with $n\sim N,\,m\sim M,\,k\sim K,\,\ell\sim L$. Then we have

[TABLE]

Especially, if $\Delta L^{1/2}\gg 1$, then

[TABLE]

Proof. See Zhai [12], Lemma 5.

Lemma 2.6

Suppose $N_{j}\geqslant 2\,(j=1,2,3,4),\,\Delta>0$. Let $\mathscr{A}_{\pm}(N_{1},N_{2},N_{3},N_{4};\Delta)$ denote the number of solutions of the following inequality

[TABLE]

with $n_{j}\sim N_{j}\,(j=1,2,3,4),\,n_{j}\in\mathbb{N}^{*}$. Then we have

[TABLE]

Proof. See Zhai [12], Lemma 3.

3 Proof of Theorem 1.1

In this section, we shall prove the theorem. We begin with the following truncated formula, which is proved by Jutila[6], i.e.,

[TABLE]

where $1\leqslant N\ll x.$

Suppose $T\geqslant 10.$ By a splitting argument, it is sufficient to prove the result in the interval $[T,2T]$. Take $y=T^{3/4}$. For any $T\leqslant x\leqslant 2T$, by the truncated formula (3.1), we get

[TABLE]

where

[TABLE]

We have

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Let

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According to the elementary formula

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we can write

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where

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[TABLE]

By (1.1) and Lemma 2.4, we get

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We now proceed to consider the contribution of $S_{4}(x)$. Applying Lemma 2.2 and (1.1), we get

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Now let us consider the contribution of $S_{2}(x)$. By the first derivative test. we have

[TABLE]

where

[TABLE]

If $M\geqslant 200L$, then $|\sqrt{n}+\sqrt{m}-\sqrt{k}-\sqrt{\ell}|\gg M^{1/2}$, so the trivial estimate yields

[TABLE]

If $L\geqslant 200M$, we can get the same estimate. So later we always suppose that $M\asymp L$. Let $\eta=\sqrt{n}+\sqrt{m}-\sqrt{k}-\sqrt{\ell}$. Write

[TABLE]

where

[TABLE]

We estimate $\mathcal{G}_{1}$ first. By Lemma 2.5, we get

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On the other hand, by Lemma 2.6, without loss of generality, we assume that $N\leqslant K\leqslant L$ and obtain

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From (3.9) and (3.10), we get

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Case 1 If $L\gg T^{1/3}$, then $T^{-3/8}L^{9/8}\gg 1,$ we get

[TABLE]

Case 2 If $L\ll T^{1/3}$, then $T^{-3/8}L^{9/8}\ll 1.$ By noting that $M\asymp L\asymp\max(N,M,K,L)$ and Lemma 2.3, we have

[TABLE]

Hence, we obtain

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Combining (3.11) and (3.12), we get

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Now, we estimate $\mathcal{G}_{2}$. By a splitting argument, we get that there exists some $\delta$ satisfying $T^{-1/2}\ll\delta\ll 1$ such that

[TABLE]

By Lemma2.5, we get

[TABLE]

On the other hand, by Lemma 2.6, without loss of generality, we assume that $N\leqslant K\leqslant L$ and obtain

[TABLE]

From (3.14) and (3.15), we get

[TABLE]

Case 1 If $\delta\gg L^{-3/2}$, then $\delta^{3/4}L^{9/8}\gg 1$, we get (recall $\delta\gg T^{-1/2}$)

[TABLE]

Case 2 If $\delta\ll L^{-3/2}$, then $\delta^{3/4}L^{9/8}\ll 1$. By Lemma 2.3, we have

[TABLE]

Therefore, we obtain (recall $\delta\gg T^{-1/2}$)

[TABLE]

Combining (3.16) and (3.17), we get

[TABLE]

For $\mathcal{G}_{3}$, by a splitting argument and Lemma 2.5 again, we get

[TABLE]

Comnining (3.7), (3.8), (3.13), (3.18) and (3.19), we get

[TABLE]

In the same way, we can prove that

[TABLE]

From (3.3)-(3.6), (3.20) and (3.21), we get

[TABLE]

which implies Theorem 1.1 immediately.

Acknowledgement

The authors would like to express the most and the greatest sincere gratitude to Professor Wenguang Zhai for his valuable advice and constant encouragement.