Power Moments of the Riesz Mean Error Term of Symmetric Square L-Function in Short Intervals

Abstract

Let $f\left(z\right)$ be a holomorphic Hecke eigenform of weight k with respect to $\mathrm{SL}(2,\mathbb{Z})$ and let $L(s,{\mathrm{sym}}^{2}f)={\displaystyle \sum _{n=1}^{\infty }}{c}_n{n}^{-s},\Re s>1$ denote the symmetric square L-function of f. In this paper, we consider the Riesz mean of the form $D_{\rho }(x;{\mathrm{sym}}^{2}f)=\frac{L(0,{\mathrm{sym}}^{2}f)}{\Gamma (\rho +1)}{x}^{\rho }+{\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)$ and derive the asymptotic formulas for ${\int }_{T-H}^{T+H}{\Delta }_{\rho }^k(x;{\mathrm{sym}}^{2}f)dx$, when $k\ge 3$.

1. Introduction

Let $S_k(\Gamma )$ be the space of all holomorphic cusp forms

$$f\left(z\right)=\sum _{n=1}^{\infty }{a}_f\left(n\right)e\left(nz\right),z\in H=\{z=x+iy:y>0\}$$

of even weight $k\ge 12$ for $\Gamma =\mathrm{SL}(2,\mathbb{Z})$; $S_k{(\Gamma )}^{+}$ denote the set of Hecke eigenforms from this space with $a_f\left(1\right)=1$. Set

$${\lambda }_f\left(n\right)=\frac{a_f\left(n\right)}{n^{\frac{k-1}{2}}}.$$

For prime p, we have

$${\lambda }_f\left(p\right)={\alpha }_f\left(p\right)+{\beta }_f\left(p\right),{\alpha }_f\left(p\right){\beta }_f\left(p\right)=1.$$

For $f\in S_k{(\Gamma )}^{+}$, we consider the symmetric square $L(s,{\mathrm{sym}}^{2}f)$ of the Hecke L-function $L(s,f)$:

$$L(s,{\mathrm{sym}}^{2}f):=\prod _p{(1-p^{-s})}^{-1}{\left(1-\frac{{\alpha }_f^2\left(p\right)}{p^{-s}}\right)}^{-1}{\left(1-\frac{{\beta }_f^2\left(p\right)}{p^{-s}}\right)}^{-1},\Re s>1$$

and have

$$L(s,{\mathrm{sym}}^{2}f)=\zeta \left(2s\right)\sum _{n=1}^{\infty }{\lambda }_f\left(n^2\right)n^{-s}=\sum _{n=1}^{\infty }{c}_n{n}^{-s}.$$

Then it is known that for any $ϵ>0$,

$$|c_n|\le d_3\left(n\right)\ll n^{ϵ},$$

(1)

where $d_3\left(n\right)$ is the number of ways to write n as a product of three factors.

It is known that the function $L(s,{\mathrm{sym}}^{2}f)$ is entire and satisfies the following functional equation:

$$\Lambda (1-s,{\mathrm{sym}}^{2}f)=\Lambda (s,{\mathrm{sym}}^{2}f),$$

where

$$\Lambda (s,{\mathrm{sym}}^{2}f):={\pi }^{-\frac{3s}{2}}\Gamma (\frac{s}{2}+\frac{1}{2})\Gamma (\frac{s}{2}+\frac{k-1}{2})\Gamma (\frac{s}{2}+\frac{k}{2})L(s,{\mathrm{sym}}^{2}f).$$

By using Landau’s theorem, Deligne’s result and properties of $L(s,{\mathrm{sym}}^{2}f)$, we deduce the bound

$$\sum _{n\le x}{c}_n\ll x^{\frac{1}{2}}{log}^{2}x.$$

Khan [1] studied the twisted first moment of $L(s,{\mathrm{sym}}^{2}f)$ at any point s on the critical line:

$$\sum _{f\in H_k}L(\frac{1}{2},{\mathrm{sym}}^{2}f)w_f^{-1}=logk+C+O\left(k^{-1/20+ϵ}\right),$$

$$\sum _{f\in H_k}L(\frac{1}{2}+it,{\mathrm{sym}}^{2}f)a_f\left(m^2\right)w_f^{-1}=\frac{\zeta (1+i2t)}{m^{1/2+it}}+\frac{\zeta (1-i2t)}{m^{1/2-it}}\frac{L_{\infty }(1/2-it)}{L_{\infty }(1/2+it)}+O\left(k^{-1/20+ϵ}\right).$$

It followed that for any point on the critical line and large enough even k, there exists an eigenform f of weight k such that $L(s,{\mathrm{sym}}^{2}f)$ is nonvanishing at that point. Hafner [2] considered the Riesz mean of the form

$$D_{\rho }(x;{\mathrm{sym}}^{2}f)=\frac{1}{\Gamma (\rho +1)}\sum _{n\le x}{(x-n)}^{\rho }{c}_n,$$

where $\rho \ge 0$ is a fixed number. Also, the Riesz mean can be represented as a sum of “residue function” and “error term”:

$$D_{\rho }(x;{\mathrm{sym}}^{2}f)=\frac{L(0,{\mathrm{sym}}^{2}f)}{\Gamma (\rho +1)}{x}^{\rho }+{\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f).$$

For $\rho =0$, we have

$$\sum _{n\le x}{c}_n=L(0,{\mathrm{sym}}^{2}f)+{\Delta }_0(x;{\mathrm{sym}}^{2}f).$$

The symmetric square L-function and the Riesz mean have been studied by many authors, for example, see [3,4,5,6,7,8]. Fomenko [9] considered ${\Delta }_{\rho }(x;{\mathrm{sym}}^{m}f)$ when $m=2$ and obtained a truncated Voronoi type formula. Wang [10] generalized Fomenko’s result to $m\ge 3$ under the hypothesis $\mathrm{Nice}(m,f)$. Voronoi type formulas examined below allow one to consider exponential sums connected with the coefficients of the Dirichlet series of automorphic L-functions. It seems that no nontrivial estimates for these sums can be obtained. It is of interest to apply the heuristic arguments proposed in [11,12,13,14] to these sums.

Ivić and Sargos [15] studied the higher moments of $\Delta \left(x\right)$, which denote the error term in the Dirichlet divisor problem. They got the asymptotic formulas for the cube and fourth integrals of $\Delta \left(x\right)$. Liu and Wang [16] studied the higher-power moments of ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)$ and got

$${\int }_1^T{\Delta }_{\rho }^{A_0}(x;{\mathrm{sym}}^{2}f)dx\ll T^{1+\frac{2\rho +1}{3}{A}_0+ϵ}$$

(2)

when $A_0=A_0\left(\rho \right)>3$. Then for any integer $3\le h<A_0$, they had the following asymptotic formula

$${\int }_1^T{\Delta }_{\rho }^h(x;{\mathrm{sym}}^{2}f)dx=\frac{6B_{\rho }(h,c)}{(3+(2\rho +1)h){\left(2\pi \right)}^{(\rho +1)h}{3}^{\frac{h}{2}}}{T}^{1+\frac{2\rho +1}{3}h}+O\left(T^{1+\frac{2\rho +1}{3}h+ϵ}(T^{-{\delta }_{\rho }(h,A_0)}+T^{-\frac{\rho }{3}})\right).$$

They also proved the results for $\rho =\frac{1}{2}$, $h=3,4,5$,

$${\int }_1^T{\Delta }_{\rho }^h(x;{\mathrm{sym}}^{2}f)dx=\frac{6B_{\frac{1}{2}}(h,c)}{(3+2h){\left(2\pi \right)}^{\frac{3}{2}h}{3}^{\frac{h}{2}}}{T}^{1+\frac{2}{3}h}+O\left(T^{1+\frac{2}{3}h-{\lambda }_{1/2}(h,6)+ϵ}\right),$$

where ${\lambda }_{1/2}(3,6)=\frac{1}{22}$, ${\lambda }_{1/2}(4,6)=\frac{1}{87}$, ${\lambda }_{1/2}(5,6)=\frac{1}{498}$.

As the vibration of ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)$ is complicable, it is natural to consider the mean value of ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)$ in short intervals. In this paper, we use the method of large value estimation and exponential sums to study the higher-power moments of the Riesz mean error term ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)$ in short intervals. Our results are as followed.

Theorem 1.

Let $k\ge 3$ be fixed integer, T and H be two real numbers, $\delta >0$ be fixed constant. Then for $T^{\frac{2}{3}+\frac{2ϵd_{\rho ,k}}{(2\rho +1)\delta }}\le H\le T$, we have

$${\int }_{T-H}^{T+H}{\Delta }_{\rho }^k(x;{\mathrm{sym}}^{2}f)dx=B_k{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx+O\left(H{T}^{\frac{2\rho +1}{3}k+ϵ}{\left(H{T}^{-\frac{2}{3}}\right)}^{-\frac{(2\rho +1)\delta }{d_{\rho ,k}}}\right),$$

where

$$B_k=2^{-k\rho -k+1}{\pi }^{-k\rho -k}{3}^{-\frac{k}{2}}{c}_k,d_{\rho ,k}=(3^{k-1}-4+k-\rho k)(k+\delta -2)+(2\rho +1)\delta$$

and $ϵ>0$.

In particular, for $\rho =\frac{1}{2}$, we can prove the following results.

Corollary 1.

For any $ϵ>0$, $k=3,4,5$, we have

$$\begin{array}{cc}\hfill & {\int }_{T-H}^{T+H}{\Delta }_{\frac{1}{2}}^3(x;{\mathrm{sym}}^{2}f)dx=B_{3}H{T}^2+O\left(H{T}^{2+ϵ}{\left(H{T}^{-\frac{2}{3}}\right)}^{-\frac{4\delta }{13+17\delta }}\right),\hfill \\ \hfill & {\int }_{T-H}^{T+H}{\Delta }_{\frac{1}{2}}^4(x;{\mathrm{sym}}^{2}f)dx=B_{4}H{T}^{\frac{8}{3}}+O\left(H{T}^{\frac{8}{3}+ϵ}{\left(H{T}^{-\frac{2}{3}}\right)}^{-\frac{2\delta }{50+27\delta }}\right),\hfill \\ \hfill & {\int }_{T-H}^{T+H}{\Delta }_{\frac{1}{2}}^5(x;{\mathrm{sym}}^{2}f)dx=B_{5}H{T}^{\frac{10}{3}}+O\left(H{T}^{\frac{10}{3}+ϵ}{\left(H{T}^{-\frac{2}{3}}\right)}^{-\frac{4\delta }{477+163\delta }}\right).\hfill \end{array}$$

Remark 1.

Please note that our results improve the results in [16] when $k=3,4,5$ and $\delta >\frac{50}{31}$.

To derive the above formulas, we start with Voronoi type formula of ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)$ to get the large value of it, where Halasz–Montgomery inequality and the estimates of exponential sums are used. For higher-power moments of ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)$, we divide the interval $[H,2H]$ into subintervals by binary method, and pick the maximal $|{\Delta }_{\rho }(x_r;{\mathrm{sym}}^{2}f)|$ in r intervals of length $V^{\rho +1}$ to get the results. Moreover, if ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)$ has an upper bound, then we can get the following theorems.

Theorem 2.

Let $3\le k<A$ be a fixed integer, $\frac{2\rho +1}{3(\rho +1)}<\theta <\frac{1}{2}$ be a real number and

$${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)\ll x^{\theta (\rho +1)}.$$

Then for any $T^{\theta [A(\rho +1)-3]+2-\frac{2\rho +1}{3}A+\sqrt{ϵ}}\le H\le T$, we have

$${\int }_{T-H}^{T+H}{\Delta }_{\rho }^k(x;{\mathrm{sym}}^{2}f)dx=B_k{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx+O\left(H{T}^{\frac{2\rho +1}{3}k-ϵ}\right).$$

(3)

Theorem 3.

Let $k\ge 3$ be a fixed integer. If ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)\ll x^{\frac{2\rho +1}{3}}$ is true, then (3) is holds for $T^{\frac{2}{3}+\sqrt{ϵ}}\le H\le T$.

2. Some Preliminary Lemmas

To prove our theorems, we need the following lemmas.

Lemma 1.

Let $x>1$, $N\le 1$. Then for any fixed ρ, $0\le \rho \le 1$, we have

$$\begin{array}{cc}\hfill {\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)=& 2^{-\rho }{\pi }^{-\rho -1}{3}^{-\frac{1}{2}}\Sigma \left(x\right)+O\left(x^{\frac{4\rho +1}{6}}{N}^{-\frac{2\rho +1}{6}+ϵ}\right)\hfill \\ \hfill & +O\left(x^{\frac{2\rho +2}{3}+ϵ}{N}^{-\frac{\rho +1}{6}}\right)+O\left(x^{ϵ}\right),\hfill \end{array}$$

(4)

where

$$\Sigma \left(x\right)=\sum _{n\le N}{c}_n{n}^{-\frac{\rho +2}{3}}{x}^{\frac{2\rho +1}{3}}cos(6\pi \sqrt[3]{nx}-\frac{\pi \rho }{2}).$$

Proof. 

See Theorem $1.1$ of Fomenko [9]. □

Lemma 2.

Let $h\ge 3$, $(i_1,\cdots ,i_h-1)\in {\{0,1\}}^{h-1}$ be such that

$$\sqrt[3]{n_1}+{(-1)}^{i_1}\sqrt[3]{n_2}+{(-1)}^{i_2}\sqrt[3]{n_3}+\cdots +{(-1)}^{i_h-1}\sqrt[3]{n_h}\ne 0.$$

Then

$$|\sqrt[3]{n_1}+{(-1)}^{i_1}\sqrt[3]{n_2}+\cdots +{(-1)}^{i_h-1}\sqrt[3]{n_h}|\gg max{(n_1,\cdots ,n_h)}^{-(3^{h-2}-3^{-1})}.$$

Proof. 

See for example Lemma $2.3$ in [17]. □

Lemma 3.

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

$${\int }_a^{b}g\left(x\right)h\left(x\right)dx\ll \left(\underset{a\le x\le b}{max}\left|g\left(x\right)\right|\right)\left(\underset{a\le u<v\le b}{max}\left|{\int }_u^{v}h\left(x\right)dx\right|\right).$$

Proof. 

See Lemma 1 in [18]. □

Lemma 4.

Suppose $k\ge 3$, $(i_1,\dots ,i_{k-1})\in {\{0,1\}}^{k-1}$, $(i_1,\dots ,i_{k-1})\ne (0,\dots ,0)$ and

$$N_1,\dots ,N_k>1,0<\Delta \ll E^{\frac{1}{3}},E=max(N_1,\dots ,N_k).$$

Let

$$\mathcal{A}=\mathcal{A}(N_1,\dots ,N_k;i_1,\dots ,i_{k-1};\Delta )$$

denote the number of solutions of the inequality

$$|\sqrt[3]{n_1}+{(-1)}^{i_1}\sqrt[3]{n_2}+\cdots +{(-1)}^{i_k-1}\sqrt[3]{n_k}|<\Delta$$

with $N_j<n_j\le 2N_j$, $1\le j\le k$. Then

$$\mathcal{A}\ll \Delta E^{-\frac{1}{3}}{N}_1\dots N_k+E^{-1}{N}_1\dots N_k.$$

Proof. 

It can be proved similarly as Lemma $2.4$ of [19], so we omit the details. □

Lemma 5.

Let S be an inner-product vector space over $\mathbb{C}$, $(a,b)$ denote the inner product in S and ${\parallel a\parallel }^2=(a,a)$. Suppose that ξ, ${\phi }_1,\cdots ,{\phi }_R$ are arbitrary vectors in S. Then

$$\sum _{l\le R}|(\xi ,{\phi }_l){|}^2\le {\parallel \xi \parallel }^2\underset{l_1\le R}{max}\sum _{l_2\le R}\left|({\phi }_{l_l},{\phi }_{l_2})\right|.$$

Proof. 

This is the well-known Halasz–Montgomery inequality (see $(A.40)$ of [20]). □

3. The Higher-Power Moments of ${\Delta }_{\mathbf{\rho }}(\mathit{x},{\mathrm{sym}}^{\mathbf{2}}\mathit{f})$ in Short Intervals

The conclusion in Theorem 1 is obvious at $\frac{T}{2}\le H\le T$. Let $H\le T/2$ and $T^{ϵ}<y\le min(T^{\frac{2\rho }{2\rho +1}},T^{\frac{1}{2}})$ be a parameter to be determined later. For any $T\le x\le 2T$, we define

$${\mathcal{R}}_1={\mathcal{R}}_1(x,y):=C\sum _{n\le y}{c}_n{n}^{-\frac{\rho +2}{3}}{x}^{\frac{2\rho +1}{3}}cos(6\pi \sqrt[3]{nx}-\frac{\pi \rho }{2}),$$

where $C=2^{-\rho }{\pi }^{-\rho -1}{3}^{-\frac{1}{2}}$;

$${\mathcal{R}}_2={\mathcal{R}}_2(x,y):={\Delta }_{\rho }(x,{\mathrm{sym}}^{2}f)-{\mathcal{R}}_1.$$

We will show that the higher-power moments of ${\mathcal{R}}_2$ is small, and thus the integral

$${\int }_{T-H}^{T+H}{\Delta }_{\rho }^k(x,{\mathrm{sym}}^{2}f)dx$$

can be approximated by ${\int }_{T-H}^{T+H}{\mathcal{R}}_1^{k}dx$.

3.1. Evaluation of the Integral ${\int }_{T-H}^{T+H}{\mathcal{R}}_1^{k}dx$

Let $k\ge 3$ be any fixed integer. For each element $\mathbf{i}=(i_1,i_2,\dots ,i_{k-1})\in {\mathbb{I}}^{k-1},\mathbb{I}=\{0,1\}$, from the elementary formula

$$cos{a}_1\cdots cos{a}_k=\frac{1}{2^{k-1}}\sum _{\mathbf{i}\in {\mathbb{I}}^{k-1}}cos(a_1+{(-1)}^{i_1}{a}_2+{(-1)}^{i_2}{a}_3+\cdots +{(-1)}^{i_{k-1}}{a}_k),$$

we have

$$\begin{array}{cc}\hfill {\mathcal{R}}_1^k=& C^k{x}^{\frac{2\rho +1}{3}k}\sum _{n_1\le y}\cdots \sum _{n_k\le y}\frac{c_{n_1}\cdots c_{n_k}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}}\prod _{j=1}^{k}cos(6\pi \sqrt[3]{n_{j}x}-\frac{\pi \rho }{2})\hfill \\ \hfill =& C^k{2}^{-k+1}{x}^{\frac{2\rho +1}{3}k}\sum _{(i_1,\dots ,i_{k-1})\in {\{0,1\}}^{k-1}}\sum _{n_1\le y}\cdots \sum _{n_k\le y}\frac{c_{n_1}\cdots c_{n_k}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}}\hfill \\ \hfill & \times cos(6\pi \sqrt[3]{x}\alpha (n_1,\dots ,n_k;i_1,\dots ,i_{k-1})-\frac{\pi \rho }{2}\beta (i_1,\dots ,i_{k-1})),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & \alpha (n_1,\dots ,n_k;i_1,\dots ,i_{k-1}):=\sqrt[3]{n_1}+{(-1)}^{i_1}\sqrt[3]{n_2}+\cdots +{(-1)}^{i_{k-1}}\sqrt[3]{n_k},\hfill \\ \hfill & \beta (i_1,\dots ,i_{k-1}):=1+{(-1)}^{i_1}+\cdots +{(-1)}^{i_{k-1}}.\hfill \end{array}$$

It follows that

$${\mathcal{R}}_1^k=C^k{2}^{-k+1}\left(S_1\left(x\right)+S_2\left(x\right)\right),$$

(5)

where

$$\begin{array}{cc}\hfill & S_1\left(x\right):=x^{\frac{2\rho +1}{3}k}\sum _{(i_1,\dots ,i_{k-1})\in {\{0,1\}}^{k-1}}cos(-\frac{\pi \rho }{2}\beta )\sum _{\genfrac{}{}{0pt}{}{n_j\le y,1\le j\le k}{\alpha =0}}\frac{c_{n_1}\cdots c_{n_k}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}},\hfill \\ \hfill & S_2\left(x\right):=x^{\frac{2\rho +1}{3}k}\sum _{(i_1,\dots ,i_{k-1})\in {\{0,1\}}^{k-1}}\sum _{\genfrac{}{}{0pt}{}{n_j\le y,1\le j\le k}{\alpha \ne 0}}\frac{c_{n_1}\cdots c_{n_k}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}}cos(6\pi \sqrt[3]{x}\alpha -\frac{\pi \rho }{2}\beta ),\hfill \\ \hfill & \alpha :=\alpha (n_1,\dots ,n_k;i_1,\dots ,i_{k-1}),\beta :=\beta (i_1,\dots ,i_{k-1}).\hfill \end{array}$$

First, we consider the contribution of $S_1\left(x\right)$. We have

$${\int }_{T-H}^{T+H}{S}_1\left(x\right)dx=\sum _{(i_1,\dots ,i_{k-1})\in {\{0,1\}}^{k-1}}cos(-\frac{\pi \rho }{2}\beta )\sum _{\genfrac{}{}{0pt}{}{n_j\le y,1\le j\le k}{\alpha =0}}\frac{c_{n_1}\cdots c_{n_k}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}}{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx.$$

By Lemma $\left(2.1\right)$ in [16] we have

$$\sum _{(i_1,\dots ,i_{k-1})\in {\{0,1\}}^{k-1}}cos(-\frac{\pi \rho }{2}\beta )\sum _{\genfrac{}{}{0pt}{}{n_j\le y,1\le j\le k}{\alpha =0}}\frac{c_{n_1}\cdots c_{n_k}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}}=c_k+O\left(y^{-\frac{2\rho +1}{3}+ϵ}\right),$$

where $c_k$ is a constant that can be expressed explicitly. So, we get

$$\begin{array}{cc}\hfill C^k{2}^{-k+1}{\int }_{T-H}^{T+H}{S}_1\left(x\right)dx=& C^k{2}^{-k+1}{c}_k{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx+O\left(H{T}^{\frac{2\rho +1}{3}k+ϵ}{y}^{-\frac{2\rho +1}{3}}\right)\hfill \\ \hfill =& B_k{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx+O\left(H{T}^{\frac{2\rho +1}{3}k+ϵ}{y}^{-\frac{2\rho +1}{3}}\right).\hfill \end{array}$$

(6)

Now we consider the contribution of $S_2\left(x\right)$. By Lemma 3 we get

$${\int }_{T-H}^{T+H}{S}_2\left(x\right)dx\ll T^{\frac{2\rho +1}{3}k+\frac{2}{3}}\sum _{(i_1,\dots ,i_{k-1})\in {\{0,1\}}^{k-1}}\sum _{\genfrac{}{}{0pt}{}{n_j\le y,1\le j\le k}{\alpha \ne 0}}\frac{c_{n_1}\cdots c_{n_k}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}\left|\alpha \right|}.$$

(7)

Let

$${\Sigma }_1=\sum _{\genfrac{}{}{0pt}{}{n_j\le y,1\le j\le k}{\alpha \ne 0}}\frac{c_{n_1}\cdots c_{n_k}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}\left|\alpha \right|},$$

by Lemma 2 we have $\left|\alpha \right|\gg y^{-(3^{k-2}-3^{-1})}$. Then by the Lemma 4, for some

$$y^{-(3^{k-2}-3^{-1})}\ll \Delta <y^{\frac{1}{3}},$$

we get

$$\begin{array}{cc}\hfill {\Sigma }_1& \ll \frac{y^{ϵ}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}\Delta }\mathcal{A}(N_1,\dots ,N_k;i_1,\dots ,i_{k-1};\Delta )\hfill \\ \hfill & \ll \frac{y^{ϵ}}{{(n_1\cdots n_k)}^{\frac{\rho +2}{3}}\Delta }(\Delta E^{-\frac{1}{3}}{N}_1\dots N_k+E^{-1}{N}_1\dots N_k)\hfill \\ \hfill & \ll y^{b_{\rho }\left(k\right)+ϵ},\hfill \end{array}$$

where $b_{\rho }\left(k\right)=3^{k-2}-\frac{4}{3}+\frac{(1-\rho )k}{3}$. So, we have

$${\int }_{T-H}^{T+H}{S}_2\left(x\right)dx\ll T^{\frac{2\rho +1}{3}k+\frac{2}{3}+ϵ}{y}^{b_{\rho }\left(k\right)}.$$

(8)

Combining (5)–(8) we obtain

Lemma 6.

For any fixed $k\ge 3$, we have

$${\int }_{T-H}^{T+H}{\mathcal{R}}_1^{k}dx=B_k{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx+O(H{T}^{\frac{2\rho +1}{3}k+ϵ}{y}^{-\frac{2\rho +1}{3}}+T^{\frac{2\rho +1}{3}k+\frac{2}{3}+ϵ}{y}^{b_{\rho }\left(k\right)}).$$

3.2. Higher-Power Moments of ${\mathcal{R}}_2$

Taking $N=T$ in (4), by the definition of ${\mathcal{R}}_2$ we have

$${\mathcal{R}}_2={\mathcal{R}}_2^{*}+O\left(T^{\frac{\rho +1}{3}+ϵ}\right),$$

(9)

where

$${\mathcal{R}}_2^{*}=C{x}^{\frac{2\rho +1}{3}}\sum _{y<n\le T}\frac{c_n}{n^{\frac{\rho +2}{3}}}cos(6\pi \sqrt[3]{nx}-\frac{\pi \rho }{2}).$$

Then we have

$${\mathcal{R}}_2\ll \left|x^{\frac{2\rho +1}{3}}\sum _{y<n\le T}\frac{c_n}{n^{\frac{\rho +2}{3}}}e\left(3\sqrt[3]{nx}\right)\right|+T^{\frac{\rho +1}{3}+ϵ},$$

(10)

which implies

$$\begin{array}{cc}\hfill {\int }_{T-H}^{T+H}{\mathcal{R}}_2^{2}dx& \ll H{T}^{\frac{2\rho +2}{3}+ϵ}+{\int }_{T-H}^{T+H}{\left|x^{\frac{2\rho +1}{3}}\sum _{y<n\le T}\frac{c_n}{n^{\frac{\rho +2}{3}}}e\left(3\sqrt[3]{nx}\right)\right|}^{2}dx\hfill \\ \hfill & \ll H{T}^{\frac{2\rho +2}{3}+ϵ}+H{T}^{\frac{4\rho +2}{3}}\sum _{y<n\le T}\frac{c_n^2}{n^{\frac{2\rho +4}{3}}}+T^{\frac{4\rho +4}{3}}\sum _{y<n<m\le T}\frac{c_n{c}_m}{{\left(nm\right)}^{\frac{\rho +2}{3}}(\sqrt[3]{n}-\sqrt[3]{m})}\hfill \\ \hfill & \ll H{T}^{\frac{4\rho +2}{3}}{y}^{-\frac{1+2\rho }{3}}.\hfill \end{array}$$

(11)

Here we have used the following estimate

$$\sum _{n\le u}{c}_n^2\ll u^{1+ϵ},\sum _{y<n<m\le T}\frac{c_n{c}_m}{{\left(nm\right)}^{\frac{\rho +2}{3}}(\sqrt[3]{n}-\sqrt[3]{m})}\ll T^{\frac{1-2\rho }{3}+ϵ}.$$

By the Lemma 6 and the Hölder inequality we have

$${\int }_{T-H}^{T+H}{\mathcal{R}}_1^{A_0}dx\ll H{T}^{\frac{2\rho +1}{3}{A}_0+ϵ},$$

(12)

where $A_0=k+\delta$. Using (2) and (12), we have

$${\int }_{T-H}^{T+H}{\mathcal{R}}_2^{A_0}dx\ll {\int }_{T-H}^{T+H}\left(|{\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f){|}^{A_0}+{\left|{\mathcal{R}}_1\right|}^{A_0}\right)dx\ll H{T}^{\frac{2\rho +1}{3}{A}_0+ϵ}.$$

(13)

For any $2<A<A_0$, by (11), (13) and Hölder inequality we have

$$\begin{array}{cc}\hfill {\int }_{T-H}^{T+H}{\mathcal{R}}_2^{A}dx& \ll {\left({\int }_{T-H}^{T+H}{\mathcal{R}}_2^{2}dx\right)}^{\frac{A_0-A}{A_0-2}}{\left({\int }_{T-H}^{T+H}{\mathcal{R}}_2^{A_0}dx\right)}^{\frac{A-2}{A_0-2}}\hfill \\ \hfill & \ll H{T}^{\frac{2\rho +1}{3}A+ϵ}{y}^{-\frac{(2\rho +1)(A_0-A)}{3(A_0-2)}}.\hfill \end{array}$$

Lemma 7.

Suppose $T^{ϵ}\le y\le T^{\frac{2\rho }{2\rho +1}}$, $2<A<A_0$, then

$${\int }_{T-H}^{T+H}{\mathcal{R}}_2^{A}dx\ll H{T}^{\frac{2\rho +1}{3}A+ϵ}{y}^{-\frac{(2\rho +1)(A_0-A)}{3(A_0-2)}}.$$

3.3. Proof of Theorem 1

Proof. 

Suppose $3\le k<A_0$ and $T^{ϵ}\le y\le T^{\frac{2\rho }{2\rho +1}}$. By the elementary formula

$${(a+b)}^k=a^k+k{a}^{k-1}b+O\left(\right|a^{k-2}{b}^2{|+|b|}^k),$$

we get

$$\begin{array}{c}\hfill {\int }_{T-H}^{T+H}{\Delta }_{\rho }^k(x,{\mathrm{sym}}^{2}f)dx={\int }_{T-H}^{T+H}{\mathcal{R}}_1^{k}dx+O({\int }_{T-H}^{T+H}|{\mathcal{R}}_1^{k-1}{\mathcal{R}}_2\left|dx\right)+O\left({\int }_{T-H}^{T+H}{\mathcal{R}}_2^{k}dx\right).\end{array}$$

(14)

By (12), Lemma 7 and Hölder inequality we get

$$\begin{array}{cc}\hfill {\int }_{T-H}^{T+H}\left|{\mathcal{R}}_1^{k-1}{\mathcal{R}}_2\right|dx& \ll {\left({\int }_{T-H}^{T+H}{\left|{\mathcal{R}}_1\right|}^{A_0}dx\right)}^{\frac{k-1}{A_0}}{\left({\int }_{T-H}^{T+H}{\left|{\mathcal{R}}_2\right|}^{\frac{A_0}{A_0-k+1}}dx\right)}^{\frac{A_0-k+1}{A_0}}\hfill \\ \hfill & \ll H{T}^{\frac{2\rho +1}{3}k+ϵ}{y}^{-\frac{(2\rho +1)(A_0-k)}{3(A_0-2)}}.\hfill \end{array}$$

(15)

Now take $y={\left(H{T}^{-\frac{2}{3}}\right)}^{\frac{1}{b_{\rho }\left(k\right)+(2\rho +1)(A_0-k)/(3A_0-6)}}$. By Lemma 6, Lemma 7 and (14)–(15) we finally get

$$\begin{array}{cc}\hfill {\int }_{T-H}^{T+H}{\Delta }_{\rho }^k(x,{\mathrm{sym}}^{2}f)dx=& B_k{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx\hfill \\ \hfill & +O(T^{\frac{2\rho +1}{3}k+\frac{2}{3}+ϵ}{y}^{b_{\rho }\left(k\right)}+H{T}^{\frac{2\rho +1}{3}k+ϵ}{y}^{-\frac{(2\rho +1)(A_0-k)}{3(A_0-2)}})\hfill \\ \hfill =& B_k{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx\hfill \\ \hfill & +O\left(H{T}^{\frac{2\rho +1}{3}k+ϵ}{\left(H{T}^{-\frac{2}{3}}\right)}^{-\frac{(2\rho +1)(A_0-k)}{b_{\rho }\left(k\right)(3A_0-6)+(2\rho +1)(A_0-k)}}\right)\hfill \\ \hfill =& B_k{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx\hfill \\ \hfill & +O\left(H{T}^{\frac{2\rho +1}{3}k+ϵ}{\left(H{T}^{-\frac{2}{3}}\right)}^{-\frac{(2\rho +1)\delta }{d_{\rho ,k}}}\right).\hfill \end{array}$$

Theorem 1 follows immediately. □

4. Proof of Theorem 2 and Theorem 3

4.1. A Large Value Estimate of ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)$

Theorem 4.

Suppose $N^{1+ϵ}\le H\le x_1<x_2<\cdots <x_k\le 2H$ satisfy $|{\Delta }_{\rho }(x_r;{\mathrm{sym}}^{2}f)|\ge V^{\rho +1}(r=1,2,\cdots ,R)$ and $|x_j-x_i|\gg V\gg H^{\frac{14}{39}}{(logH)}^{\frac{30}{13}}(i\ne j)$. Then we have

$$R\ll H^2{V}^{-4}{log}^{6}H+H^{\frac{13}{2}}{V}^{-\frac{59}{4}}{log}^{29}H.$$

Proof. 

Suppose that $H_0>V$ is a parameter to be determined later. Let I be any subinterval of $[H,2H]$ of length not exceeding $H_0$ and let $G=I\cap \{x_1,x_2,\cdots ,x_R\}$. Without loss of generality, we may assume that $G=\{x_1,x_2,\cdots ,x_{R_0}\}$. Suppose $J=\left[\frac{(2+2ϵ)logH-3logV}{log2}\right]$. For any $x_r\in \{x_1,x_2,\cdots ,x_R\}$, we apply Formula (4) with $x=x_r$ and $M=2^{J+1}\asymp H^{2+2ϵ}{V}^{-3}$ to get

$$\begin{array}{cc}\hfill {\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)& \ll H^{\frac{2\rho +1}{3}}\left|\sum _{n\le M}\frac{c_n}{n^{\frac{\rho +2}{3}}}e(\pm 3{\left(n{x}_r\right)}^{\frac{1}{3}})\right|+H^{\frac{1-2\rho }{3}ϵ}{V}^{\rho +1}\hfill \\ \hfill & \ll H^{\frac{2\rho +1}{3}}\left|\sum _{j=0}^J\sum _{n\sim 2^j}\frac{c_n}{n^{\frac{\rho +2}{3}}}e(\pm 3{\left(n{x}_r\right)}^{\frac{1}{3}})\right|,\hfill \end{array}$$

if we notice that $|{\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)|\gg V^{\rho +1}$. Squaring, summing over the set G and then using the Cauchy inequality, we get that

$$\begin{array}{cc}\hfill \sum _{r\le R_0}{\Delta }_{\rho }^2(x;{\mathrm{sym}}^{2}f)& \ll H^{\frac{4\rho +2}{3}}\sum _{r\le R_0}{\left|\sum _{j=0}^J\sum _{n\sim 2^j}\frac{c_n}{n^{\frac{\rho +2}{3}}}e(\pm 3{\left(n{x}_r\right)}^{\frac{1}{3}})\right|}^2\hfill \\ \hfill & \ll H^{\frac{4\rho +2}{3}}logH\sum _{r\le R_0}\sum _{j=0}^J{\left|\sum _{n\sim 2^j}\frac{c_n}{n^{\frac{\rho +2}{3}}}e(\pm 3{\left(n{x}_r\right)}^{\frac{1}{3}})\right|}^2\hfill \\ \hfill & \ll H^{\frac{4\rho +2}{3}}{log}^{2}H\sum _{r\le R_0}{\left|\sum _{n\sim 2^{j_0}}\frac{c_n}{n^{\frac{\rho +2}{3}}}e(\pm 3{\left(n{x}_r\right)}^{\frac{1}{3}})\right|}^2\hfill \end{array}$$

(16)

for some $0\le j_0\le J$.

Let $M_0=2^{j_0}$. Take $\xi \left(d\right)={\left\{{\xi }_n\left(d\right)\right\}}_{n=1}^{\infty }$ with ${\xi }_n\left(d\right)=c_n{n}^{-\frac{\rho +2}{3}}$ for $n\sim M_0$ and zero otherwise. Take ${\phi }_r^{\pm }\left(d\right)={\left\{{\phi }_{r,n}^{\pm }\left(d\right)\right\}}_{n=1}^{\infty }$ with ${\phi }_{r,n}^{\pm }\left(d\right)=e(\mp 3{\left(n{x}_r\right)}^{\frac{1}{3}})$ for $n\sim M_0$ and zero otherwise. Then

$$\begin{array}{cc}\hfill & (\xi \left(d\right),{\phi }_r^{\pm }\left(d\right))=\sum _{n\sim M_0}\frac{c_n}{n^{\frac{\rho +2}{3}}}e(\pm 3{\left(n{x}_r\right)}^{\frac{1}{3}}),\hfill \\ \hfill & ({\phi }_{r_1}^{\pm }\left(d\right),{\phi }_{r_2}^{\pm }\left(d\right))=\sum _{n\sim M_0}e(\mp 3n^{\frac{1}{3}}(x_{r_1}^{\frac{1}{3}}-x_{r_2}^{\frac{1}{3}})),\hfill \\ \hfill & {\parallel \xi \left(d\right)\parallel }^2=\sum _{n\sim M_0}\frac{c_n^2}{n^{\frac{2\rho +4}{3}}}\ll M_0^{-\frac{1+2\rho }{3}}{log}^3{M}_0,\hfill \end{array}$$

where we have used the estimate

$$\sum _{n\sim M_0}{c}_n^2\ll M_0{log}^3{M}_0.$$

By (16) and Lemma 5 we get

$$\begin{array}{cc}\hfill R_0{V}^{2\rho +2}& \ll H^{\frac{4\rho +2}{3}}{log}^{2}H\sum _{r\le R_0}{\left|\sum _{n\sim M_0}\frac{c_n}{n^{\frac{\rho +2}{3}}}e(\pm 3{\left(n{x}_r\right)}^{\frac{1}{3}})\right|}^2\hfill \\ \hfill & \ll H^{\frac{4\rho +2}{3}}{M}_0^{-\frac{1+2\rho }{3}}{log}^{5}H\underset{r_1\le R_0}{max}\sum _{r_2\le R_0}\left|\sum _{n\sim M_0}e(\mp 3n^{\frac{1}{3}}(x_{r_1}^{\frac{1}{3}}-x_{r_2}^{\frac{1}{3}}))\right|\hfill \\ \hfill & \ll H^{\frac{4\rho +2}{3}}{M}_0^{-\frac{1+2\rho }{3}}{log}^{5}H\left(M_0+\underset{r_1\le R_0}{max}\sum _{\begin{array}{c}{r}_2\le R_0\\ r_1\ne r_2\end{array}}\left|\sum _{n\sim M_0}e(\mp 3n^{\frac{1}{3}}(x_{r_1}^{\frac{1}{3}}-x_{r_2}^{\frac{1}{3}}))\right|\right).\hfill \end{array}$$

(17)

Following the calculation on [21], we get, by the Kuzmin–Landau inequality and the exponent pair $(\frac{4}{18},\frac{11}{18})$,

$$\begin{array}{cc}\hfill \sum _{n\sim M_0}e\left(3n^{\frac{1}{3}}(x_{r_1}^{\frac{1}{3}}-x_{r_2}^{\frac{1}{3}})\right)& \ll M_0^{\frac{2}{3}}|x_{r_1}^{\frac{1}{3}}-x_{r_2}^{\frac{1}{3}}{|}^{-1}+M_0^{-\frac{2}{3}·\frac{4}{18}}{|x_{r_1}^{\frac{1}{3}}-x_{r_2}^{\frac{1}{3}}|}^{\frac{4}{18}}{M}_0^{\frac{11}{18}}\hfill \\ \hfill & \ll \frac{{\left(M_{0}H\right)}^{\frac{2}{3}}}{|x_{r_1}-x_{r_2}|}+\frac{|x_{r_1}-x_{r_2}{|}^{\frac{4}{18}}}{{\left(M_{0}H\right)}^{\frac{4}{27}}}{M}_0^{\frac{11}{18}}\hfill \\ \hfill & \ll \frac{{\left(M_{0}H\right)}^{\frac{2}{3}}}{|x_{r_1}-x_{r_2}|}+H^{-\frac{4}{27}}{H}_0^{\frac{2}{9}}{M}_0^{\frac{25}{54}},\hfill \end{array}$$

where we have used the mean value theorem and the estimate $|x_{r_1}-x_{r_2}|\le H_0$. Substituting this estimate into (17), we get

$$\begin{array}{cc}\hfill R_0{V}^{2\rho +2}& \ll H^{\frac{4\rho +2}{3}}{M}_0^{-\frac{1+2\rho }{3}}{log}^{5}H\left(M_0+\underset{r_1\le R_0}{max}\sum _{\begin{array}{c}{r}_2\le R_0\\ r_1\ne r_2\end{array}}\left(\frac{{\left(M_{0}H\right)}^{\frac{2}{3}}}{|x_{r_1}-x_{r_2}|}+H^{-\frac{4}{27}}{H}_0^{\frac{2}{9}}{M}_0^{\frac{25}{54}}\right)\right)\hfill \\ \hfill & \ll H^{\frac{4\rho +2}{3}}{M}_0^{-\frac{1+2\rho }{3}}{log}^{5}H\left(M_0+M_0^{\frac{2}{3}}{H}^{\frac{2}{3}}{V}^{-1}logH+H^{-\frac{4}{27}}{H}_0^{\frac{2}{9}}{M}_0^{\frac{25}{54}}{R}_0\right)\hfill \\ \hfill & \ll H^{\frac{4\rho +2}{3}}{M}_0^{\frac{2-2\rho }{3}}{log}^{5}H+M_0^{\frac{1-2\rho }{3}}{H}^{\frac{4+4\rho }{3}}{V}^{-1}{log}^{6}H+H^{\frac{14}{27}+\frac{4\rho }{3}}{H}_0^{\frac{2}{9}}{M}_0^{\frac{7}{54}-\frac{2\rho }{3}}{R}_0{log}^{5}H\hfill \\ \hfill & \ll M_0^{\frac{1-2\rho }{3}}{H}^{\frac{4+4\rho }{3}}{V}^{-1}{log}^{6}H+H^{\frac{14}{27}+\frac{4\rho }{3}}{H}_0^{\frac{2}{9}}{M}_0^{\frac{7}{54}-\frac{2\rho }{3}}{R}_0{log}^{5}H,\hfill \end{array}$$

where we use the facts that $\left\{x_r\right\}$ is V-spaced and $M_0\ll H^{2+2ϵ}{V}^{-3}$. Take

$$H_0=c{H}^{-\frac{7}{2}}{V}^{\frac{43}{4}}{log}^{-\frac{45}{2}}H$$

for some sufficiently small constant c such that we can get for this $H_0$ that

$$R_0{V}^{2\rho +2}\ll H^2{V}^{-2+2\rho }{log}^{6}H,$$

i.e.,

$$R_0\ll H^2{V}^{-4}{log}^{6}H.$$

It is easy to check that $H_0\gg V$ if $V\gg H^{\frac{14}{39}}{(logH)}^{\frac{30}{13}}$.

Now we divide the interval $[H,2H]$ into $O(1+\frac{H}{H_0})$ subintervals of length not exceeding $H_0$. In each interval of this type, the number of $x_r$’s is at most $O\left(H^2{V}^{-4}{log}^{6}H\right)$. So, we have

$$\begin{array}{cc}\hfill R& \ll R_0\left(1+\frac{H}{H_0}\right)\hfill \\ \hfill & \ll H^2{V}^{-4}{log}^{6}H+H^3{V}^{-4}{H}_0^{-1}{log}^{6}H\hfill \\ \hfill & \ll H^2{V}^{-4}{log}^{6}H+H^{\frac{13}{2}}{V}^{-\frac{59}{4}}{log}^{29}H.\hfill \end{array}$$

□

4.2. Proof of Theorem 2

By (4) Liu and Wang [16] obtained that

$${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)\ll x^{\frac{\rho +1}{2}}.$$

Let $\frac{2\rho +1}{3(\rho +1)}<\theta <\frac{1}{2}$ be a real number, such that ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)\ll x^{\theta (\rho +1)}$. Suppose $T-H\le x_1<x_2<\cdots <x_k\le T+H$ satisfy $|{\Delta }_{\rho }(x_r;{\mathrm{sym}}^{2}f)|\ge V^{\rho +1}(r=1,2,\cdots ,R_V)$ and $|x_j-x_i|\gg V,(i\ne j)$, $T^{\frac{2\rho +1}{3(\rho +1)}}\ll V\ll T^{\theta }$, by Theorem 4 we have

$$R\ll T^2{V}^{-4}{log}^{6}T+H{T}^{\frac{11}{2}}{V}^{-\frac{59}{4}}{log}^{29}T.$$

(18)

An argument similar to $\left(13.70\right)$ of [20] proved that

$${\int }_{T-H}^{T+H}{\Delta }_{\rho }^A(x,{\mathrm{sym}}^{2}f)dx\ll H{T}^{\frac{2\rho +1}{3}A}+\sum _{V}V\sum _{r\le R_V}{\left|{\Delta }_{\rho }^k(x_r,{\mathrm{sym}}^{2}f)\right|}^A,$$

(19)

where

$$T^{\frac{2\rho +1}{3(\rho +1)}}\ll V\ll T^{\theta }$$

and

$$V^{\rho +1}\le \left|{\Delta }_{\rho }(x_r;{\mathrm{sym}}^{2}f)\right|\le 2V^{\rho +1}(r=1,2,\cdots ,R_V).$$

Therefore, by (18) we have

$$\begin{array}{cc}\hfill V& \sum _{r\le R_V}{\left|{\Delta }_{\rho }^k(x_r,{\mathrm{sym}}^{2}f)\right|}^A\hfill \\ \hfill & \ll R_V{V}^{A(\rho +1)+1}\ll T^{2+ϵ}{V}^{A(\rho +1)-3}+H{T}^{\frac{11}{2}+ϵ}{V}^{A(\rho +1)-\frac{55}{4}}\hfill \\ \hfill & \ll T^{\theta \left[A\right(\rho +1)-3]+2+ϵ}+H{T}^{\frac{11}{2}+\frac{2\rho +1}{3}A-\frac{55(2\rho +1)}{12(\rho +1)}+ϵ}.\hfill \end{array}$$

(20)

By (19) and (20) we can get

$${\int }_{T-H}^{T+H}{\Delta }_{\rho }^A(x,{\mathrm{sym}}^{2}f)dx\ll H{T}^{\frac{2\rho +1}{3}A+ϵ}+T^{\theta \left[A\right(\rho +1)-3]+2+ϵ},$$

where $\frac{1}{4}\le \rho \le 1$. When $2<A<\frac{9\theta -3}{3\theta (\rho +1)-2\rho -1}$, if $H\ge T^{\theta [A(\rho +1)-3]+2-\frac{2\rho +1}{3}A}$, then we have

$${\int }_{T-H}^{T+H}{\Delta }_{\rho }^A(x,{\mathrm{sym}}^{2}f)dx\ll H{T}^{\frac{2\rho +1}{3}A+ϵ}.$$

(21)

Please note that for every $2<A<\frac{9\theta -3}{3\theta (\rho +1)-2\rho -1}$, taking $\delta =\frac{3d_{\rho ,k}\sqrt{ϵ}}{2(2\rho +1)}$ in Theorem 1, we can get the result of Theorem 2.

4.3. Proof of Theorem 3

Suppose ${\Delta }_{\rho }(x;{\mathrm{sym}}^{2}f)\ll x^{\frac{2\rho +1}{3}}$, we can obtain the following estimate

$${\int }_{T-H}^{T+H}{\Delta }_{\rho }^{k+\delta }(x,{\mathrm{sym}}^{2}f)dx\ll H{T}^{\frac{2\rho +1}{3}(k+\delta )+{ϵ}^{\prime }},$$

(22)

where ${ϵ}^{\prime }=(k+1)ϵ$. By Theorem 1 we can get

$${\int }_{T-H}^{T+H}{\Delta }_{\rho }^k(x;{\mathrm{sym}}^{2}f)dx=B_k{\int }_{T-H}^{T+H}{x}^{\frac{2\rho +1}{3}k}dx+O\left(H{T}^{\frac{2\rho +1}{3}k+{ϵ}^{\prime }}{\left(H{T}^{-\frac{2}{3}}\right)}^{-\frac{(2\rho +1)\delta }{d_{\rho ,k}}}\right)$$

when $T^{\frac{2}{3}+\frac{2{ϵ}^{\prime }{d}_{\rho ,k}}{(2\rho +1)\delta }}\le H\le T$.

Let $\delta =\frac{3d_{\rho ,k}(k+1)\sqrt{ϵ}}{2(2\rho +1)}$ in Theorem 1, then we get Theorem 3.