The third-power moment of the Riesz mean error term of symmetric square $L$-function

1.   Introduction

Let $\mathbb{H}$ be the upper half plane, $i.e.$, $\mathbb{H} = \{z = x+iy|\ x, y\in \mathbb{R}, \text{and} \ y > 0\}$. Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even weight $k\geq 2$ for the full modular group $\Gamma = SL(2, Z)$. Then for $f\in S_{k}(\Gamma)$ and $z\in \mathbb{H}$, we have the Fourier expansion

where $a_{f}(n)$ denotes the $n$-th Fourier coefficient, and $e(x) = e^{2\pi i x}$. Set $\lambda_{f}(n) = a_{f}(n)n^{-\frac{k-1}{2}}$, then $\lambda_{f}(n)$ is multiplicative. The analytic properties of $\lambda_{f}(n)$ were studied by many authors (see ^{[5,7,8,9,10,12,13,14,16,20,22,23,24,25]}). The symmetric square $L$-function attached to $f$ can be written as

for $\Re s > 1$. And it can be continued to an entire function on the whole complex plane. Then it is known that for any $\epsilon > 0$,

where $d_{3}(n)$ is the number of ways to write $n$ as a product of three factors.

Hafner ^[3] considered the Riesz mean of the form

where $\rho\geq0$ is a fixed number and $\sum'$ indicates that $c_n$ is replaced by $c_n/2$ if $\rho = 0$. Also, the Riesz mean can be represented as a sum of "residue function" and "error term":

Then let $\Delta_{\rho}(x; sym^{2}f)$ be the error term of the Riesz mean of the symmetric square $L$-function. Hafner ^[3] gave that for $\rho = 0$, one has

The symmetric square $L$-function and the Riesz mean error term have been studied by many authors, for example, see ^{[1,4,6,11,15,18,21,25]}. Fomenko ^[2] considered $\Delta_{\rho}(x; \text{sym}^{m}f)$ when $m = 2$ and obtained a truncated Voronoï type formula, and we give it in Lemma 2.1. Let $N = x^{\frac{1}{2}}$ in Lemma 2.1, one obtain

Using the Voronöi formula, Fomenko got

where

Wang ^[19] generalized the truncated Voronöi type formula to the case $m\geq 3$ under the hypothesis $\text{Nice}(m, f)$:

where

with

Liu and Wang ^[15] studied the higher-power moments of $\Delta_{\rho}(x; sym^{2}f)$. If exists a real number $A_0 = A_0(\rho) > 3$ such that

then for any integer $3\leq h < A_0$, they had the following asymptotic formula

where

with

and

with

and $H_{0}$ is the least even integer such that $n \geqslant A_{0}$. They also proved the result for $\rho = \frac{1}{2}$, $h = 3, 4, 5$,

where $\lambda_{1/2}(3, 6) = \frac{1}{22}$, $\lambda_{1/2}(4, 6) = \frac{1}{87}$, $\lambda_{1/2}(5, 6) = \frac{1}{498}$. Zhang, Han and Zhang ^[26] studied the power moment of the Riesz mean error term in short intervals. For $k\geq3$, $T^{\frac{2}{3}+\frac{2\epsilon d_{\rho, k}}{(2\rho+1)\delta}}\leq H\leq T$ and $\epsilon > 0$, they got

where

Their results improved the results in ^[15] when $\rho = \frac{1}{2}$, $k = 3, 4, 5$ and $\delta > \frac{50}{31}$.

In this paper, we will use the Voronöi type formula for $\Delta_{\rho}(x; sym^{2}f)$ to study the third-power moment estimates of $\Delta_{\rho}(x; sym^{2}f)$ and obtain the following theorem.

Theorem 1. For any $f\in S_{k}(\Gamma)^{+}$, $x > 1$ and $\frac{\sqrt{612}}{18}-1 < \rho < \frac{2}{3}$, we have

where

Particularly, for $\rho = \frac{1}{2}$, we can get the following result.

Corollary 2. Let $\rho = \frac{1}{2}$. For any $\epsilon > 0$, we have

where

Remark. Note that our results improve the results in ^[15].

2.   Some preliminary lemmas

To prove our theorem, we need the following lemmas.

Lemma 2.1. Let $x > 1$, $N\geq1$. Then for any fixed $\rho$, $0\leq \rho \leq 1$, we have

where

Proof. See Theorem $1.1$ in ^[2].

Lemma 2.2. Let $h\geq3$, $(i_1, \cdots, i_{h-1})\in \{0, 1\}^{h-1}$ such that

Then

Proof. See for example Lemma $2.3$ in ^[21].

Lemma 2.3. If $g(x)$ and $h(x)$ are continuous real-valued functions of $x$ and $g(x)$ is monotonic, then

Proof. See the Lemma $1$ in ^[17].

Lemma 2.4. Suppose $k\geq3$, $(i_1, \ldots, i_{k-1})\in\{0, 1\}^{k-1}$, $(i_1, \ldots, i_{k-1})\neq(0, \ldots, 0)$ and

Let

denote the number of solutions of the inequality

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

Proof. It can be proved similarly as Lemma $2.4$ of ^[21]. Without loss of generality, suppose $E = N_{k}$. If $n_{1}, \ldots, n_{k}$ satisfy (2.2), then

for some $|\theta| < 1$. We can get

Hence for fixed $n_{1}, \ldots, n_{k-1}$, the number of $n_k$ is $\ll 1+\Delta N_{k}^{\frac{2}{3}}$ and thus

3.   The third-power moment of $\Delta_\rho(x, sym^{2}f)$

Throughout this paper, $n$, $m$ and $k$ denote natural numbers. The constants implied in the symbols $\ll$ and $O$ may depend on $\epsilon$.

By (2.1), we know that to establish the asymptotic formula in our theorem, we shall prove, for $H\geq1$, that

and take $N = H^{\frac{6\rho+8}{(8+3\rho)(\rho+1)}}$,

Then, since $(a+b)^{3} = a^{3}+O\left(|b|a^2+|b|^3\right)$, it follows from (2.1) that

by (3.1) and (3.2). Adding this for $H = T/2, T/2^2, \ldots,$ we see that the asymptotic formula in our theorem follows.

We shall now give the details of the proof of (3.2). The proof of (3.1), which we shall omit, employs similar arguments and is simpler.

For simplicity, put

and $r = 0$ otherwise. Taking the third power of both sides of $\Sigma(x)$ and for each element ${\bf{i}} = (i_1, i_2)\in\mathbb{I}^{2}, \mathbb{I} = \{0, 1\}$, using the elementary formula

we can write

where

The main term in the right-hand side of (3.2) comes from $S_0(x)$. Indeed, integrating both sides of (3.6) with respect to $x$ over the interval $(H, 2H)$, we have

For natural numbers $n$, $m$ and $k$, the condition $\sqrt[3]{n}+\sqrt[3]{m} = \sqrt[3]{k}$ holds if and only if $n$, $m$ and $k$ all have the same cube-free part $h$, such that $n = \alpha^3h$, $m = \beta^3h$, $k = \gamma^3h$ and $\alpha+\beta = \gamma$. Hence

Since $c_{uv}\leq c_{u}c_{v}$ and $c_{w}\ll w^{\epsilon}$, we have

Thus

Substituting this into (3.9), we have

This yields the main term in (3.2) and hence that in the formula in our theorem.

We now proceed to show that the contributions from $S_1(x)$ and $S_2(x)$ are bounded by $H^{\frac{6\rho^{2}+21\rho+16}{8+3\rho}+\epsilon}$. Applying the Lemma 2.3 we find that for any real numbers $p(\neq0)$ and $q$,

Let us examine $S_2(x)$ first. Integrating both sides of (3.8) and then using (3.11) and (3.4), we have

Next we consider $S_1(x)$. For convenience, let

Using the trivial bound

where $|\Delta|\leq H^{-\frac{8}{24+9\rho}}$ and applying (3.11) when $|\Delta| > H^{-\frac{8}{24+9\rho}}$, we deduce from (3.7) that

Now we consider first the sum $W_1$. For given $n\leq m\leq N$ there is most one natural number $k$ for which $|\Delta|\leq H^{-\frac{8}{24+9\rho}}$, since $\Delta = \sqrt[3]{n}+\sqrt[3]{m}-\sqrt[3]{k}$ and $\left|\Delta\sqrt[3]{N^{2}}\right|\leq H^{-\frac{8}{24+9\rho}}\sqrt[3]{N^{2}} = o(1)$. Such $k$, if it exists, must be greater than $m$. By Lemma 2.2, $0 < |\Delta|\leq H^{-\frac{8}{24+9\rho}}$ implies $m^{-\frac{8}{3}}\ll H^{-\frac{8}{24+9\rho}}$, so $m\gg H^{\frac{1}{8+3\rho}}$. Hence, by (3.4),

Similarly, in the sum $W_2$, we have $k = \left(\sqrt[3]{n}+\sqrt[3]{m}\right)^{3}+O\left(|\Delta|\sqrt[3]{m^{2}}\right) > m$ and there are $\ll 1+|\Delta|\sqrt[3]{m^{2}}$ such $k$. Hence

Collecting these estimates we find that

Combining this with (3.10) and (3.12), we deduce from (3.2) the formula (3.5). Then according to (3.3), our theorem is proved.

Acknowledgments

Zhang is supported by the National Natural Science Foundation of China [Grant No. 11771256]. Yan is supported by the National Natural Science Foundation of China [Grant No. 11801327] and Natural Science Foundation of Shandong Province [Grant No. ZR201709280100].

Conflict of interest

The authors declare no conflict of interest.