The short interval results for power moments of the Riesz mean error term

1.   Introduction

Let $\varphi(z)$ be a holomorphic form of weight $\kappa$ with respect to the full modular group $SL_2(\mathbb{Z})$ and denote by $a(n)$ the $n$-th Fourier coefficient of $\varphi(z)$. We assume that $\varphi(z)$ is normalized such that $a(1) = 1$ and $T(n)\varphi = a(n)\varphi$ for every $n \in \mathbb{N}$, where $T(n)$ is the Hecke operator of order $n$. Let $c_n$ be the convolution function defined by

In 1974, Deligne ^[1] proved the estimate $|a(n)|\leq n^{\frac{\kappa-1}{2}}d(n)$, where $d(n)$ is the Dirichlet divisor function, which implies $c_n\ll_\varepsilon n^\varepsilon$. Here and in what follows, $\varepsilon$ denotes an arbitrarily small positive number which is not necessarily the same at each occurrence. The classical Rankin-Selberg problem is to estimate the upper bound of the error term

where $C$ is an explicit constant. In 1939, Rankin ^[2] proved that

which was stated by Selberg ^[3] again without proof. However, no improvement of (1.2) has been obtained after Rankin and Selberg. In ^[4], Ivić obtained that $\Delta(x; \varphi) = \Omega_\pm(x^{3/8})$ and conjectured that $\Delta(x; \varphi) = O(x^{3/8+\varepsilon})$.

Ivić, Matsumoto and Tanigawa ^[5] considered the Riesz mean of the type

for any fixed $\rho\geq0$ and define the error term $\Delta_\rho(c; \varphi)$ by

where

where $Z(s)$ can be continued to the whole plane and the integral being taken over a fundamental domain $\mathfrak{F}$ of $SL_2(\mathbb{Z})$. They considered the relation between $\Delta(x; \varphi)$ and $\Delta_1(x; \varphi)$ and proved that $\Delta(x; \varphi) = O(x^{\alpha/2})$ if $\Delta_1(x; \varphi) = O(x^\alpha)$ holds for some $\alpha\geq0$. They also proved that

and

Since this kind sums play a very important role in the study of analytic number theory, many number theorists and scholars have obtained a series of meaningful research results (for example see ^{[6,7,8,9,11,12,13]}, etc.). In particular, in ^[9], Tanigawa, Zhai and Zhang studied the third, fourth and fifth power moments of $\Delta_1(x; \varphi)$ and proved that

where

In this paper we shall prove that the $k$-th power moment of $\Delta_1(x; \varphi)$ in short intervals for $k = 3, 4, 5$, the theorem is as follows.

Theorem 1. Let $k\geq3$ be a fixed integer. For any sufficiently small $\varepsilon > 0$, let $\delta_k: = (k-1)\left(4^{k-1}+\frac{k-10}{2}\right)+3$, $0 < \delta < 1$ be a fixed constant, which satisfies $\frac{8}{3}\delta_k\varepsilon < \delta$, $T$ and $H$ are two large positive real number, which satisfies

and $T^{3/4+2\delta_k\varepsilon/(3\delta)}\leq H\leq T$. Then, we have

where

Remark 1. If we take $H = T$, $\delta$ is larger (for example $\delta = \frac{1}{2}$), then Theorem 1 implies asymptotic formula (1.4).

As corollaries, we have the following Theorems 2 and 3. Theorem 3 implies the best possible result.

Theorem 2. Suppose $3\leq k\leq5$, $1/8 < \theta < 1/5$ is a real number. Let $\Delta_1(x; \varphi)\ll x^{\theta+1}$. Then, we have asymptotic formula

when $T^{1+(k-2)\theta-k/8+\sqrt{\varepsilon}}\leq H\leq T$.

Corollary 1. For $3\leq k\leq5$, if $T^{(k-2)/5+1-k/8+\sqrt{\varepsilon}}\leq H\leq T$, asymptotic formula (1.7) is true.

Theorem 3. Suppose $k\geq3$ be a any fixed integer and conjecture $\Delta_1(x; \varphi) = O(x^{9/8+\varepsilon})$ is true. Then, asymptotic formula (1.7) is true if $T^{3/4+\sqrt{\varepsilon}/2}\leq H\leq T$.

Remark 2. By variable substitution, it is easy to see that

here

If the conjecture $\Delta_1(x; \varphi) = O(X^{9/8+\varepsilon})$ is true and $H = T^{3/4+\sqrt{\varepsilon}/2}$, then we have $H'\asymp T^{\sqrt{\varepsilon}/2}$. Thus, Theorem 3 contains the integral $\int_{T-G}^{T+G}\Delta_1^k(x^4;\varphi)dx$ has asymptotic formula for $G = T^{\sqrt{\varepsilon}/2}$. Thus, the constant $3/4$ in Theorem 3 is probably the best.

2.   Some Preliminary Lemmas

Lemma 1. Suppose $x > 1$ is a real number. For $1\ll N\ll x^2$ a parameter we have

where

Proof. This is [9, Lemma 2.1].

Lemma 2. Suppose $k\geq3$, $(i_1, \cdots, i_{k-1})\in\{0, 1\}^{k-1}$ such that

Then, we have

Proof. This is [9, Lemma 2.3].

Lemma 3. If $f(x)$ and $g(x)$ are continuous real-valued functions of $x$ and $f(x)$ is monotonic, then we have

Proof. This follows from the second mean value theorem.

For any real numbers $p(\neq0)$ and $q$, by using this lemma we can obtain

Lemma 4. Suppose $k\geq3$, $(i_1, \cdots, i_{k-1})\in\{0, 1\}^{k-1}$, $(i_1, \cdots, i_{k-1})\neq(0, \cdots, 0)$, $N_1, \cdots, N_k > 1$, $0 < \Delta\ll E^{1/4}$, $E = \max(N_1, \cdots, N_k)$. Let $\mathcal{A}$ denote the number of solutions of the inequality

with $N_j < n_j\leq2N_j$, $1\leq j\leq k$, where

Then, we have

Proof. The proof of this lemma is similar to the proof of [10, Lemma 2.4]. Suppose $E = N_k$. If $(n_1, \cdots, n_k)$ satisfies (2.2), then for some $|\theta| < 1$, we can obtain

Thus, we have

Therefore, for fixed $(n_1, \cdots, n_{k-1})$, the number of $n_k$ is $\ll1+\Delta N_k^{3/4}$ and so

3.   Proof of Theorem 1

If $T/2\leq H\leq T$, it is easily to get Theorem 1. Suppose $H\leq\frac{T}{2}$ and $y$ is a parameter such that $T^\varepsilon < y\leq T^{1/3}$. For any $T\leq x\leq2T$, we define

We will prove that the higher-power moment of $\mathcal{R}_1$ is small, so the integral $\int_{T-H}^{T+H}\Delta_1^k(x; \varphi)dx$ can be well approximated by $\int_{T-H}^{T+H}\mathcal{R}^kdx$, which is easy to evaluate.

3.1.   Evaluation of the integral $\int_{T-H}^{T+H}\mathcal{R}^kdx$

First, by the elementary formula

we can write

where

Therefore, we have

where

Consider $S_3(x, k)$. We have

By (4.3) in ^[9], we know that

Now we consider $S_4(x, k)$. By the first derivative of van der Corput method, one has

For fixed $(i_1, \cdots, i_{k-1})\in\{0, 1\}^{k-1}$, we write

If $(i_1, \cdots, i_{k-1}) = (0, \cdots, 0)$, then we have

For $(i_1, \cdots, i_{k-1})\neq(0, \cdots, 0)$, by a splitting argument we deduce that there exist a collection of numbers $1 < N_1, \cdots, N_k < y$ such that

Suppose $N_1\leq\cdots\leq N_k\leq y$. According to Lemma 2, we know that $|\alpha|\gg N_k^{-(4^{k-2}-4^{-1})}$. For some $N_k^{-(4^{k-2}-4^{-1})}\ll \Delta < y^{1/4}$, by using Lemma 4 we can get

where $b(k) = 4^{k-2}+\frac{k-10}{8}$. Therefore we have

According to (3.1)-(3.6), we have

3.2.   Higher-power moments of $\mathcal{R}_1$

From the definition of $K_0$ ($K_0: = \min\{n\in\mathbb{N}:n\geq A_0, 2|n \}$ in ^[9] we write

Suppose that $y\leq(HT^{-3/4})^{1/b(K_0)}$. Let $N = T$ in the formula (2.1) of Lemma 1. Then, we can obtain

Thus, we have

From (3.7) we can obtain that

By Hölder's inequality and the above formula we have

for $A_0 = k+\delta$, here $0 < \delta < 1$ is a fixed constant. From (1.5) and (3.9) we get

For any $2 < A < A_0$, by (3.8), (3.10) and Hölder inequality we have

Thus we have

Lemma 5. Suppose $T^\varepsilon\leq y\leq (HT^{-3/4})^{1/b(K_0)}$, $2 < A < A_0$. Then, we have

3.3.   Proof of Theorem 1

Suppose $3\leq k < A_0$. By the elementary formula $(a+b)^k-a^k\ll |b|^k+|a^{k-1}b|$ we have

By (3.9), Lemma 5 and Hölder inequality we have

Taking $y = (HT^{-3/4})^{\frac{1}{b(K_0)+\frac{3\delta}{4(k+\delta-2)}}}$. From (3.13), (3.14), (3.7) and Lemma 5, we can obtain

From this we can easily get Theorem 1.

4.   Proof of Theorem 2

In this section, we will give the proof of Theorem 2. Let $N = T^{3/5}$. By (2.1) we can obtain

Suppose $\frac{1}{8} < \theta < \frac{1}{5}$ be a fixed real number. Let $\Delta_1(x; \varphi)\ll x^{\theta+1}$.

Suppose $T-H\leq x_1 < \cdots < x_R\leq T+H$ satisfies $|x_r-x_s|\geq V\ (r\neq s\leq R)$, $T^{1/8}\ll V\ll T^\theta$ and $|\Delta_1(x_r; \varphi)|\gg VT\ (r = 1, \cdots, R)$. Dividing interval $[T-H, T+H]$ into the subinterval of length not exceeding $T_0\ (T_0\geq V)$. Let $R_0$ denotes the number of $x_r$ which exists in the subinterval of length not exceeding $T_0$. Then, we have

Tanigawa, Zhai and Zhang ^[9] proved that

if $T_0 = V^{22}T^{-2}\mathcal{L}^{-34}$, $R_0\ll T^{1+4\varepsilon}V^{-2}$.

Now we consider $\int_{T-H}^{T+H}|\Delta_1(x; \varphi)|^Adx$, where $2 < A < 24$ be a fixed real number. According to ^[9], we have

where $T^{1/8}\leq V = 2^m\leq T^\theta$, $VT < |\Delta_1(x_r; \varphi)|\leq2VT\ (r = 1, \cdots, R_V)$ and $|x_r-x_s|\geq V$ for $r\neq s\leq R\leq R_V$. By (4.1) we have

Combining (4.2) and (4.3), we get

Now suppose $2 < A < 2\theta/(\theta-1/8)$. If $H\geq T^{1+\theta(A-2)-A/8}$, then we have

We notice that $2\theta/(\theta-1/8) = 16/3 = 5.33\cdots$, if $\theta = 1/5$. In Theorem 1, taking $\delta = 4/3\delta_k\sqrt{\varepsilon}$ for $3\leq k\leq5$, we can get Theorem 2.

5.   Proof of Theorem 3

Suppose $\Delta_1(x; \varphi)\ll x^{9/8+\varepsilon}$. Then,

where $\varepsilon' = (k+1)\varepsilon$. From Theorem 1, we can get asymptotic formula

for $T^{\frac{2\delta_k\varepsilon'}{3\delta}+\frac{3}{4}}\leq H\leq T$. Taking $\delta = \frac{4}{3}\delta_k(k+1)\sqrt{\varepsilon}$ in Theorem 1, we can get Theorem 3.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by National Natural Science Foundation of China (grant nos. 12171286).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.