On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Abstract

Let ${\mathbb{P}}_c\left(x\right)=\{p\le x|p,\left[p^c\right]\mathrm{are}\mathrm{primes}\},c\in {\mathbb{R}}^{+}\setminus \mathbb{N}$ and ${\lambda }_{sy{m}^{2}f}\left(n\right)$ be the n-th Fourier coefficient associated with the symmetric square L-function $L(s,sy{m}^{2}f)$. For any $A>0$, we prove that the mean value of ${\lambda }_{sy{m}^{2}f}\left(n\right)$ over ${\mathbb{P}}_c\left(x\right)$ is $\ll x{log}^{-A-2}x$ for almost all $c\in \left(\epsilon ,(\sqrt{5}+3)/8-\epsilon \right)$ in the sense of Lebesgue measure. Furthermore, it holds for all $c\in (0,1)$ under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for ${\lambda }_f^2\left(n\right)$ over ${\mathbb{P}}_c\left(x\right)$ is ${\sum }_{p,qprimep\le x,q=\left[p^c\right]}{\lambda }_f^2\left(p\right)=\frac{x}{c{log}^{2}x}(1+o\left(1\right)),$ for almost all $c\in \left(\epsilon ,(\sqrt{5}+3)/8-\epsilon \right)$, where ${\lambda }_f\left(n\right)$ is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

1. Introduction

Let k be an even positive integer, f be a holomorphic cusp form of weight k for the full modular group and ${\lambda }_f\left(n\right)$ be the normalized n-th Fourier coefficient of f, i.e.,

$$f\left(z\right)=\sum _{n=1}^{\infty }{\lambda }_f\left(n\right)n^{\frac{k-1}{2}}e\left(nz\right).$$

If we assume that f is an eigenform of all the Hecke operators, then f can be normalized such that ${\lambda }_f\left(1\right)=1$ and ${\lambda }_f\left(n\right)$ is real. We define the Hecke L-function associated to f for $\Re s>1$ by

$$L(s,f)=\sum _{n=1}^{\infty }\frac{{\lambda }_f\left(n\right)}{n^s}=\prod _p{(1-{\lambda }_f\left(p\right)p^{-s}+p^{-2s})}^{-1}.$$

For any prime p and all integers $\nu \ge 0$, we have

$${\lambda }_f\left(p^{\nu }\right)={\alpha }_f{\left(p\right)}^{\nu }+{\alpha }_f{\left(p\right)}^{\nu -1}{\beta }_f\left(p\right)+\cdots +{\beta }_f{\left(p\right)}^{\nu },$$

where ${\alpha }_f\left(p\right),{\beta }_f\left(p\right)$ are the local parameters of $L(s,f)$ at prime p, satisfying

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

Then we have

$${\lambda }_f^2\left(p\right)=1+{\lambda }_f\left(p^2\right).$$

Deligne [1,2] proved Ramanujan–Petersson conjecture, i.e.,

$$|{\lambda }_f\left(n\right)|\le d\left(n\right)\ll n^{\epsilon }$$

for all $n\ge 1$, where $d\left(n\right)={\sum }_{d|n}1$.

In order to detect the sign changes of ${\lambda }_f\left(n\right)$, many authors have studied the mean value of ${\lambda }_f\left(n\right)$ and obtained some good results. For example, see [1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. In addition, the sums of ${\lambda }_f\left(n\right)$ over primes have also been studied. It is known that (see for example Section 5.6 of Iwaniec and Kowalski [23]) there exists a constant $C>0$ such that

$$\sum _{p\le N}{\lambda }_f\left(p\right){\ll }_{f}Nexp(-C\sqrt{logN}).$$

(1)

The upper bound of (1) may reach $N^{\frac{1}{2}+\epsilon }$, assuming the Riemann Hypothesis. Furthermore, we can establish that

$$\sum _{p\le N}{\lambda }_f^2\left(p\right)\sim \frac{N}{logN},\mathrm{as}N\to \infty ,$$

by using the analytic properties of the Rankin–Selberg L-function $L(s,f\otimes \overline{f})$.

Another interesting question considered by many authors is the mean value of ${\lambda }_f\left(n\right)$ over certain sets of primes. For example, Baier and Zhao [24] studied the distribution of ${\lambda }_f\left(n\right)$ at Piatetski–Shapiro primes by considering estimates of exponential sum involving Hecke eigenvalue. Moreover, they conjectured that, for $1<c<1+\theta$ with some suitable $\theta >0$, there exists a constant $c_f>0$ such that

$$\sum _{\genfrac{}{}{0pt}{}{n\le N}{\left[n^c\right]\in \mathbb{R}}}{\lambda }_f^2\left(\left[n^c\right]\right)\sim c_f\frac{N}{clogN},\mathrm{as}N\to \infty .$$

Furthermore, we can define Piatetski–Shapiro prime twins if $p,q$ are primes and $q=\left[p^c\right]$, for any fixed $c\in {\mathbb{R}}^{+}\setminus \mathbb{N}$. Balog [25] and Dufner [26] proved that

$$\sum _{\genfrac{}{}{0pt}{}{p,qprime}{p\le x,q=\left[p^c\right]}}1=\frac{x}{c{log}^{2}x}(1+o\left(1\right)),x\to \infty ,$$

(2)

for almost all $c\in (0,1)$ in the sense of Lebesgue measure. Furthermore, assuming the Riemann Hypothesis of automorphic L-function $L(s,f)$ is true, they found that (2) holds for all $c\in (0,1)$. Furthermore, Zhang and Zhai [27] studied the mean value of ${\lambda }_f\left(n\right)$ over Piatetski–Shapiro prime twins.

Motivated by the above results, we are interested in the distribution of ${\lambda }_f^2\left(n\right)$ at Piatetski–Shapiro prime twins. For the form f, we know the $L(s,sy{m}^{2}f)$ is an L-function for some $GL(3,\mathbb{Z})$ automorphic representation, which is often called the symmetric-square lift of f. The n-th Fourier coefficient of $L(s,sy{m}^{2}f)$ satisfies

$${\lambda }_{sy{m}^{2}f}\left(n\right)=\sum _{m{l}^2=n}{\lambda }_f\left(m^2\right).$$

The symmetric square L-function associated to f is defined by

$$L(s,sy{m}^{2}f)=\prod _p\prod _{0\le j\le 2}{\left(1-\frac{{\alpha }_f{\left(p\right)}^{2-j}{\beta }_f{\left(p\right)}^j}{p^s}\right)}^{-1}=\sum _{n=1}^{\infty }\frac{{\lambda }_{sy{m}^{2}f}\left(n\right)}{n^s}$$

in the half-plane $\Re s>1$. Then, for all $n\ge 1$, ${\lambda }_{sy{m}^{2}f}\left(n\right)$ is also multiplicative, real and

$$|{\lambda }_{sy{m}^{2}f}\left(n\right)|\le d_3\left(n\right),$$

where $d_3\left(n\right)={\sum }_{m|n}d\left(m\right)$. Furthermore, for all primes p, we have

$${\lambda }_{sy{m}^{2}f}\left(p\right)={\lambda }_f\left(p^2\right).$$

Many authors studied the mean value of ${\lambda }_{sy{m}^{2}f}\left(n\right)$. For example, see References [28,29,30,31,32,33,34,35,36]. In this paper, we consider the mean value of Fourier coefficients of symmetric square L-function over Piatetski–Shapiro prime twins and obtain the following results, which imply a result on the distribution of ${\lambda }_f^2\left(n\right)$ at Piatetski–Shapiro prime twins.

Theorem 1.

For almost all $c\in \left(\epsilon ,(\sqrt{5}+3)/8-\epsilon \right)$ and any $A>0$, we have

$$\sum _{\genfrac{}{}{0pt}{}{p,qprime}{p\le x,q=\left[p^c\right]}}{\lambda }_{sy{m}^{2}f}\left(p\right)\ll \frac{x}{{log}^{A+2}x}.$$

(3)

Theorem 2.

Assuming the Riemann Hypothesis of symmetric square L-function is true, (3) holds for all $c\in (0,1)$.

Corollary 1.

For almost all $c\in \left(\epsilon ,(\sqrt{5}+3)/8-\epsilon \right)$, we have

$$\sum _{\genfrac{}{}{0pt}{}{p,qprime}{p\le x,q=\left[p^c\right]}}{\lambda }_f^2\left(p\right)=\frac{x}{c{log}^{2}x}(1+o\left(1\right)),x\to \infty .$$

(4)

Proof. 

The result follows easily from Theorem 1 and (2), if we notice that ${\lambda }_f^2\left(p\right)=1+{\lambda }_f\left(p^2\right)$.      ☐

Corollary 2.

Assuming the Riemann Hypothesis of symmetric square L-function is true, (4) holds for all $c\in (0,1)$.

Proof. 

The result follows from Theorem 2 and (2).      ☐

Notation. 

Throughout the paper, ε always denotes a sufficiently small positive constant. Let ${\delta }_1\left(\epsilon \right)$ be sufficiently small and depend on ε. We write $f\left(x\right)\ll g\left(x\right)$, or $f\left(x\right)=O\left(g\right(x\left)\right)$, to mean that $\left|f\right(x\left)\right|\le Cg\left(x\right)$. Let $\rho =\sigma +i\eta$ be the nontrivial zero of the symmetric square L-function $L(s,sy{m}^{2}f)$. As usual, $\mathrm{\Lambda }\left(n\right)$ is the von Mangoldt function.

2. Auxiliary Lemmas

Lemma 1.

Let η run through a countable set of reals, with $c\left(\eta \right)$ arbitrary complex such that $S\left(t\right)={\sum }_{\eta }c\left(\eta \right)e\left(\eta t\right)$ is absolutely convergent. Let $\alpha \in \mathbb{R},\Theta \in (0,1),\delta =\Theta /\alpha$. Then

$$\begin{array}{c}\hfill {\int }_{-\alpha }^{\alpha }{\left|S\left(t\right)\right|}^{2}dt{\ll }_{\Theta }{\int }_{-\infty }^{\infty }|{\delta }^{-1}\sum _{t\le \eta \le t+\delta }c\left(\eta \right){|}^{2}dt.\end{array}$$

Proof. 

This lemma is Lemma 1 of Gallagher [37].      ☐

Lemma 2.

Let $T\ge T_0$, where $T_0$ is a sufficiently large real number. The

$$\begin{array}{c}\hfill {\int }_T^{2T}{\left|L(\alpha +it,sy{m}^{2}f)\right|}^{2}dt\ll \left\{\begin{array}{cc}(10+k+T){(log(10+k+T))}^{17},\hfill & if2/3\le \alpha \le 1,\hfill \\ {(10+k+T)}^{3(1-\alpha )}{(log(10+k+T))}^{17},\hfill & if1/3\le \alpha \le 2/3,\hfill \\ {(10+k+T)}^{3(1-2\alpha )+1}{(log(10+k+T))}^{17},\hfill & if0\le \alpha \le 1/3.\hfill \end{array}\right.\end{array}$$

Proof. 

This lemma is Lemma $3.1$ of Lü [33].      ☐

Lemma 3.

For $1/2\le \sigma \le 1,T\ge 3$, define

$$\begin{array}{c}\hfill N(\sigma ,T)=♯\{\varrho =\beta +i\omega :L(\varrho ,sy{m}^{2}f)=0,\sigma \le \beta \le 1,|\omega |\le T\}.\end{array}$$

Then we have

$$\begin{array}{c}\hfill N(\sigma ,T)\ll T^{\frac{5(1-\sigma )}{3-2\sigma }+\epsilon },if1/2\le \sigma \le 3/4,\end{array}$$

and

$$\begin{array}{c}\hfill N(\sigma ,T)\ll T^{3(1-\sigma )+\epsilon },if3/4<\sigma \le 1.\end{array}$$

Proof. 

From Lemma 2, we have

$${\int }_T^{2T}|L(1/2+it,sy{m}^{2}f){|}^{2}dt\ll T^{3/2+\epsilon }.$$

(5)

This combined with Theorem 1.1, 1.2 of Ye and Zhang [38] and [39] gives this lemma.      ☐

Lemma 4.

For any $u\ge 2$, $T\ge 2$, we have

$$\begin{array}{c}\hfill \sum _{n<u}\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right)=-\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T}}\frac{u^{\rho }}{\rho }+O\left({log}^{3}u+\frac{u{log}^3\left(uT\right)}{T}\right).\end{array}$$

Proof. 

See, for example, Iwaniec and Kowalski [23]. For convenience of calculation, we reduce the summation range of n from $n\le u$ to $n<u$. The contribution of $n=u$ is $O\left({log}^{3}u\right)$, in view of $|{\lambda }_{sy{m}^{2}f}\left(p^h\right)|\le d_3\left(p^h\right)\ll {log}^{2}u$.      ☐

Lemma 5.

Let $L(s,f)$ be an L-function of degree k such that Rankin–Selberg convolutions $L(s,f\otimes f)$ and $L(s,f\otimes \overline{f})$ exist, and the latter has a simple pole at $s=\alpha +it=1$ while the former is entire if $f\ne \overline{f}$. Suppose that the ramified primes $|{\alpha }_f\left(p\right)|\le p/2$. There exists an absolute constant $c>0$ such that $L(s,f)$ has no zeros in the region

$$\alpha \ge 1-\frac{c}{k^{4}log\left(\mathfrak{q}\left(f\right)\right|t|+3)},$$

where $\mathfrak{q}(f,s)$ is the analytic conductor and $\mathfrak{q}\left(f\right)=\mathfrak{q}(f,0)$.

Proof. 

This lemma is Theorem 5.10 of Iwaniec and Kowalski [23].      ☐

Lemma 6.

Let x, $T\in {\mathbb{R}}^{+}$, $x\to \infty$ and

$$S:=\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T}}{x}^{\sigma },$$

we have

$$\begin{array}{c}\hfill S\ll \left\{\begin{array}{cc}{x}^{1/2}{T}^{5/4+\epsilon },\hfill & ifT\ge x^{4/5};\hfill \\ x^{3/2}{T}^{5/2+\epsilon }{e}^{-\sqrt{5logMlogT}},\hfill & if{x}^{9/20}\le T<x^{4/5};\hfill \\ x^{3/4}{T}^{5/6+\epsilon }+x^{1-\beta }{T}^{3\beta +\epsilon },\hfill & ifT<x^{9/20}.\hfill \end{array}\right.\end{array}$$

Proof. 

Note that

$$S={\int }_{1/2}^{1-\beta }{x}^{\sigma }{d}_{\sigma }N(\sigma ,T),$$

where $\beta =\beta \left(T\right)$ is the width of the zero-free region of the symmetric square L-function in Lemma 5. Using integration by parts and Lemma 3, we obtain

$$\begin{array}{cc}\hfill S& {\ll }_l{x}^{1/2}T+{\int }_{1/2}^{3/4}exp\left(f_1\left(\sigma \right)\right)d\sigma +{\int }_{3/4}^{1-\beta }exp\left(f_2\left(\sigma \right)\right)d\sigma \hfill \\ \hfill & :=x^{1/2}T+S_1+S_2,\hfill \end{array}$$

(6)

where ${\ll }_l$ is fulfilled apart from a fixed number of log-factors,

$$\begin{array}{c}\hfill f_1\left(\sigma \right)=\frac{5(1-\sigma )}{3-2\sigma }logT+\sigma logx+\epsilon logT\end{array}$$

and

$$\begin{array}{c}\hfill f_2\left(\sigma \right)=3(1-\sigma )logT+\sigma logx+\epsilon logT.\end{array}$$

We estimate $S_1$ first. The first and second derivatives of $f_1\left(\sigma \right)$ are

$$f_1^{{}^{\prime }}\left(\sigma \right)=-\frac{5logT}{{(3-2\sigma )}^2}+logx$$

and

$$f_1^{{}^{″}}\left(\sigma \right)=-\frac{20logT}{{(3-2\sigma )}^3}<0,\mathrm{if}\sigma <3/2.$$

Then $f_1\left(\sigma \right)$ is a concave function as $\sigma <3/2$. Let $f_1^{{}^{\prime }}\left({\sigma }_1\right)=0$, we have ${\sigma }_1=\frac{3}{2}-\frac{1}{2}\sqrt{\frac{5logT}{logx}}$. So we have the following three cases.

Case 1. When $T\ge x^{4/5}$. In this case, we have ${\sigma }_1\le 1/2$ and $f_1\left(\sigma \right)$ is monotonically decreasing in $[1/2,3/4]$. Hence

$$\underset{\sigma \in [1/2,3/4]}{max}{f}_1\left(\sigma \right)=f_1\left(1/2\right)=(5logT)/4+(logx)/2+\epsilon logT.$$

Case 2. When $x^{9/20}\le T<x^{4/5}$. In this case, the function $f_1\left(\sigma \right)$ takes the extreme value at ${\sigma }_1$, which gives

$$\underset{\sigma \in [1/2,3/4]}{max}{f}_1\left(\sigma \right)=f_1\left({\sigma }_1\right)=(5logT)/2+(3logx)/2-\sqrt{5logTlogx}+\epsilon logT.$$

Case 3. When $T<x^{9/20}$. In this case, we have ${\sigma }_1>3/4$ and $f_1\left(\sigma \right)$ is monotonically increasing in $[1/2,3/4]$. Hence,

$$\underset{\sigma \in [1/2,3/4]}{max}{f}_1\left(\sigma \right)=f_1\left(3/4\right)=(5logT)/6+(3logx)/4+\epsilon logT.$$

Combining all the above cases, we have

$$\begin{array}{c}\hfill S_1={\int }_{\frac{1}{2}}^{\frac{3}{4}}exp\left(f_1\left(\sigma \right)\right)d\sigma \ll \left\{\begin{array}{cc}{x}^{1/2}{T}^{5/4+\epsilon },\hfill & \mathrm{if}T\ge x^{4/5};\hfill \\ x^{3/2}{T}^{5/2+\epsilon }{e}^{-\sqrt{5logxlogT}},\hfill & \mathrm{if}{x}^{9/20}\le T<x^{4/5};\hfill \\ x^{3/4}{T}^{5/6+\epsilon },\hfill & \mathrm{if}T<x^{9/20}.\hfill \end{array}\right.\end{array}$$

(7)

Next, we need to estimate $S_2$. It is easy to see that $f_2\left(\sigma \right)$ is linear function in $[3/4,1-\beta ]$, hence

$$\begin{array}{cc}\hfill f_2\left(\sigma \right)& \ll max\left(f_2\left(3/4\right),f_2(1-\beta )\right)\hfill \\ \hfill & =max\left((3logT)/4+(3logx)/4,3\beta logT+(1-\beta )logx\right)+\epsilon logT.\hfill \end{array}$$

(8)

From (6), (7) and (8), we obtain

$$\begin{array}{c}\hfill S{\ll }_l\left\{\begin{array}{cc}{x}^{1/2}T+x^{1/2}{T}^{5/4+\epsilon }+x^{3/4}{T}^{3/4+\epsilon },\hfill & \mathrm{if}T\ge x^{4/5};\hfill \\ x^{1/2}T+x^{3/2}{T}^{5/2+\epsilon }{e}^{-\sqrt{5logxlogT}}+x^{3/4}{T}^{3/4+\epsilon },\hfill & \mathrm{if}{x}^{9/20}\le T<x^{4/5};\hfill \\ x^{1/2}T+x^{3/4}{T}^{5/6+\epsilon }+x^{1-\beta }{T}^{3\beta +\epsilon },\hfill & \mathrm{if}T<x^{9/20},\hfill \end{array}\right.\end{array}$$

which proves this lemma.      ☐

Lemma 7.

Let $x_0:=2$, $x_{v+1}:=x_v+x_v{log}^{-2}{x}_v$. Let ${\gamma }_0$ be a constant and

$$D({\gamma }_0,x_v,T)={\int }_{{\gamma }_0}^{{\gamma }_0+{log}^{-2}{x}_v}{\left|\sum _{x_v^{1/\gamma }<m\le x_{v+1}^{1/\gamma }}\mathrm{\Lambda }\left(m\right)\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{T/2<\left|\eta \right|\le T}}\frac{{(m+1)}^{\gamma \rho }-m^{\gamma \rho }}{\rho }\right|}^{2}d\gamma .$$

If $T>x_v^{1/{\gamma }_0-\delta }$, we have

$$D({\gamma }_0,x_v,T)\ll x_v^{2/{\gamma }_0}{T}^{-3}{\left(\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{T/2<\left|\eta \right|\le T}}{x}_v^{\sigma }\right)}^2.$$

(9)

If $T\le x_v^{1/{\gamma }_0-\delta }$, we have

$$D({\gamma }_0,x_v,T)\ll x_v^{2/{\gamma }_0}{T}^{-3}{\left(\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{T/2<\left|\eta \right|\le T}}{x}_v^{\sigma }\right)}^2{\left(T{x}_v^{-1/{\gamma }_0}\right)}^2.$$

(10)

Proof. 

This lemma follows from (13) and (16) of Dufner [26].      ☐

3. Proof of Theorem 2

In this section, we write $\gamma =1/c$. Then we have

$${\Pi }_{2,c}\left(x\right):=\sum _{\genfrac{}{}{0pt}{}{p,qprime}{p\le x,q=\left[p^c\right]}}{\lambda }_{sy{m}^{2}f}\left(p\right)=\sum _{\genfrac{}{}{0pt}{}{p,qprime}{p\le x,q\le p^c<(q+1)}}{\lambda }_{sy{m}^{2}f}\left(p\right)=\sum _{q\le x^c}\sum _{\genfrac{}{}{0pt}{}{p\le x}{q^{\gamma }\le p<{(q+1)}^{\gamma }}}{\lambda }_{sy{m}^{2}f}\left(p\right).$$

We split the summation range of q into two parts: $q\le x^c-1$ and $x^c-1<q\le x^c$ to get

$$\begin{array}{cc}\hfill {\Pi }_{2,c}\left(x\right)& =\sum _{q\le x^c-1}\sum _{q^{\gamma }\le p<{(q+1)}^{\gamma }}{\lambda }_{sy{m}^{2}f}\left(p\right)+\sum _{x^c-1<q\le x^c}\sum _{q^{\gamma }\le p\le x}{\lambda }_{sy{m}^{2}f}\left(p\right)\hfill \\ \hfill & =\sum _{q\le x^c}\sum _{q^{\gamma }\le p<{(q+1)}^{\gamma }}{\lambda }_{sy{m}^{2}f}\left(p\right)-\sum _{x^c-1<q\le x^c}\sum _{x<p<{(q+1)}^{\gamma }}{\lambda }_{sy{m}^{2}f}\left(p\right)\hfill \\ \hfill & :=E_1-E_2.\hfill \end{array}$$

(11)

For $E_2$, there is at most one prime $q_x$ satisfying $x^c-1<q_x\le x^c$, hence

$$E_2\ll \sum _{q_x^{\gamma }<p\le {(q_x+1)}^{\gamma }}\left|{\lambda }_{sy{m}^{2}f}\left(p\right)\right|\ll {(q_x+1)}^{\gamma }-q_x^{\gamma }\ll x^{1-c},$$

(12)

if we notice that $|{\lambda }_{sy{m}^{2}f}\left(p\right)|\le d_3\left(p\right)\ll 1$.

For $E_1$, by the definition of the von Mangoldt function, we have

$$E_1=\sum _{q\le x^c}\sum _{q^{\gamma }\le n<{(q+1)}^{\gamma }}\frac{\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right)}{logn}+O\left(\sum _{q\le x^c}\sum _{\genfrac{}{}{0pt}{}{q^{\gamma }\le n=p^h<{(q+1)}^{\gamma }}{h\ge 2}}\frac{{\lambda }_{sy{m}^{2}f}\left(p^h\right)logp}{log{p}^h}\right).$$

(13)

The error term of the above formula contributes

$$\begin{array}{cc}\hfill & \ll \sum _{q\le x^c}\sum _{\genfrac{}{}{0pt}{}{q^{\gamma }\le p^h<{(q+1)}^{\gamma }}{h\ge 2}}\left|{\lambda }_{sy{m}^{2}f}\left(p^h\right)\right|\hfill \\ \hfill & =\sum _{q\le x^c}\sum _{q^{\gamma }\le p^2<{(q+1)}^{\gamma }}|{\lambda }_{sy{m}^{2}f}\left(p^2\right)|+\sum _{q\le x^c}\sum _{\genfrac{}{}{0pt}{}{q^{\gamma }\le p^h<{(q+1)}^{\gamma }}{h\ge 3}}\left|{\lambda }_{sy{m}^{2}f}\left(p^h\right)\right|\hfill \\ \hfill & \ll \sum _{q\le x^c}\sum _{q^{\gamma }\le n^2<{(q+1)}^{\gamma }}1+\sum _{q\le x^c}\sum _{\genfrac{}{}{0pt}{}{q^{\gamma }\le n^h<{(q+1)}^{\gamma }}{h\ge 3}}{log}^{2}x\hfill \\ \hfill & \ll \sum _{q\le x^c}\left({(q+1)}^{\gamma /2}-q^{\gamma /2}\right)+\sum _{q\le x^c}\left({(q+1)}^{\gamma /3}-q^{\gamma /3}\right){log}^{3}x\hfill \\ \hfill & \ll \sum _{q\le x^c}{q}^{\gamma /2-1}+\sum _{q\le x^c}{q}^{\gamma /3-1}{log}^{3}x\ll x^{1/2}.\hfill \end{array}$$

We use the same method to deal with the main term of (13) and get

$$E_1=\sum _{2\le m\le x^c}\frac{\mathrm{\Lambda }\left(m\right)}{logm}\sum _{m^{\gamma }\le n<{(m+1)}^{\gamma }}\frac{\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right)}{logn}+O\left(x^{1-c/2}\right)+O\left(x^{1/2}\right),$$

(14)

where the first O-term comes from

$$\begin{array}{cc}\hfill & \sum _{\genfrac{}{}{0pt}{}{m=q^h\le x^c}{h\ge 2}}\frac{\mathrm{\Lambda }\left(m\right)}{logm}\sum _{m^{\gamma }\le n<{(m+1)}^{\gamma }}\frac{\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right)}{logn}\hfill \\ \hfill & =\sum _{\genfrac{}{}{0pt}{}{m=q^h\le x^c}{h\ge 2}}\frac{logq}{log{q}^h}\sum _{m^{\gamma }\le p^r<{(m+1)}^{\gamma }}\frac{{\lambda }_{sy{m}^{2}f}\left(p^r\right)logp}{log{p}^r}\hfill \\ \hfill & \ll \sum _{\genfrac{}{}{0pt}{}{m=q^h\le x^c}{h\ge 2}}\sum _{m^{\gamma }\le p^r<{(m+1)}^{\gamma }}\left|{\lambda }_{sy{m}^{2}f}\left(p^r\right)\right|\ll x^{1-c/2}.\hfill \end{array}$$

Let

$$W\left(x\right):=\sum _{2\le m\le x^c}\frac{\mathrm{\Lambda }\left(m\right)}{logm}\sum _{m^{\gamma }\le n<{(m+1)}^{\gamma }}\frac{\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right)}{logn}.$$

By the range of n and Taylor’s formula, we have

$$logn=\gamma logm+O\left(m^{-1}\right)$$

and

$$\frac{1}{logn}=\frac{1}{\gamma logm}+O\left(\frac{1}{m{log}^{2}m}\right).$$

Then

$$W\left(x\right)={\gamma }^{-1}\sum _{2\le m\le x^c}\frac{\mathrm{\Lambda }\left(m\right)}{{log}^{2}m}\sum _{m^{\gamma }\le n<{(m+1)}^{\gamma }}\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right)+O\left(x^{1-c}{log}^{3}x\right),$$

(15)

where the O-term comes from

$$\begin{array}{cc}\hfill & \sum _{2\le m\le x^c}\frac{\mathrm{\Lambda }\left(m\right)}{m{log}^{3}m}\sum _{m^{\gamma }\le n<{(m+1)}^{\gamma }}\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right)\hfill \\ \hfill \ll & {log}^{3}x\sum _{2\le m\le x^c}{m}^{\gamma -2}{log}^{-3}m\hfill \\ \hfill \ll & x^{1-c}{log}^{3}x\sum _{2\le m\le x^c}\frac{1}{m{log}^{3}m}\hfill \\ \hfill \ll & x^{1-c}{log}^{3}x.\hfill \end{array}$$

From (14), (15) and partial summation, we obtain

$$E_1\ll {log}^{-2}x·\underset{x_0\le x}{max}\left|{\mathrm{\Psi }}_{2,c}\left(x_0\right)\right|+x^{1-c}{log}^{3}x+x^{1-c/2}+x^{1/2},$$

(16)

where

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_{2,c}\left(x\right)=\sum _{m\le x^c}\mathrm{\Lambda }\left(m\right)\sum _{m^{\gamma }\le n<{(m+1)}^{\gamma }}\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right).\end{array}$$

To get (3), it suffices to prove that

$$\begin{array}{c}\hfill |{\mathrm{\Psi }}_{2,c}\left(x\right)|\ll x{log}^{-A}x,\mathrm{for}\mathrm{any}A>0.\end{array}$$

(17)

The inequality (17) will be proved from a variant of (17) for short intervals. Let $x_0:=2$, $x_{v+1}:=x_v+x_v{log}^{-2}{x}_v$, we can see that the inequality

$$|\Delta {\mathrm{\Psi }}_{2,c}\left(x_v\right)|=\left|\sum _{x_v^c<m\le x_{v+1}^c}\mathrm{\Lambda }\left(m\right)\sum _{m^{\gamma }\le n<{(m+1)}^{\gamma }}\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right)\right|\ll x_v{log}^{-A-2}{x}_v$$

(18)

implies (17). We use Lemma 4 and get

$$\sum _{m^{\gamma }\le n<{(m+1)}^{\gamma }}\mathrm{\Lambda }\left(n\right){\lambda }_{sy{m}^{2}f}\left(n\right)=-\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T_v}}\frac{{(m+1)}^{\gamma \rho }-m^{\gamma \rho }}{\rho }+O\left({log}^{3}m+\frac{{(m+1)}^{\gamma }{log}^3\left(mT\right)}{T}\right).$$

(19)

Taking $T_v=x_v{log}^{-2A-3}{x}_v$, then we obtain

$$|\Delta {\mathrm{\Psi }}_{2,c}\left(x_v\right)|\ll |\mathrm{\Phi }(\gamma ,x_v,T_v)|+x_v^c{log}^{2A+4}{x}_v,$$

(20)

where

$$\mathrm{\Phi }(\gamma ,x_v,T_v)=\sum _{x_v^c<m\le x_{v+1}^c}\mathrm{\Lambda }\left(m\right)\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T_v}}\frac{{(m+1)}^{\gamma \rho }-m^{\gamma \rho }}{\rho }.$$

(21)

Under the Riemann Hypothesis, we have

$$\begin{array}{cc}\hfill \mathrm{\Phi }(\gamma ,x_v,T_v)& =\sum _{x_v^c<m\le x_{v+1}^c}\mathrm{\Lambda }\left(m\right){\int }_{m^{\gamma }}^{{(m+1)}^{\gamma }}\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T_v}}{t}^{-1/2+i\eta }dt\hfill \\ \hfill & =\sum _{x_v^c<m\le x_{v+1}^c}\mathrm{\Lambda }\left(m\right){\int }_{m^{\gamma }}^{{(m+1)}^{\gamma }}\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T_v}}{e}^{(-1/2+i\eta )logt}dt.\hfill \end{array}$$

Making the change of variables $u=logt$, we deduce that

$$\mathrm{\Phi }(\gamma ,x_v,T_v)=\sum _{x_v^c<m\le x_{v+1}^c}\mathrm{\Lambda }\left(m\right){\int }_{\gamma logm}^{\gamma log(m+1)}{e}^{u/2}\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T_v}}{e}^{i\eta u}du.$$

Using the Cauchy–Schwarz inequality twice, we get

$$\begin{array}{cc}\hfill |\mathrm{\Phi }(\gamma ,x_v,T_v){|}^2& \ll \sum _{x_v^c<m\le x_{v+1}^c}{\mathrm{\Lambda }}^2\left(m\right)\sum _{x_v^c<m\le x_{v+1}^c}{\left|{\int }_{\gamma logm}^{\gamma log(m+1)}{e}^{u/2}\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T_v}}{e}^{i\eta u}du\right|}^2\hfill \\ \hfill & \ll \sum _{x_v^c<m\le x_{v+1}^c}{\mathrm{\Lambda }}^2\left(m\right)\sum _{x_v^c<m\le x_{v+1}^c}{m}^{\gamma -1}{\int }_{\gamma logm}^{\gamma log(m+1)}{\left|\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T_v}}{e}^{i\eta u}\right|}^{2}du\hfill \\ \hfill & \ll x_v{log}^{-1}{x}_v·{\int }_{log{x}_v}^{\gamma +log{x}_{v+1}}{\left|\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{\left|\eta \right|\le T_v}}{e}^{i\eta u}\right|}^{2}du.\hfill \end{array}$$

(22)

Making the change of variables $t=u{\left(2\pi \right)}^{-1}-(\gamma +log{x}_{v+1}+log{x}_v){\left(4\pi \right)}^{-1}$, we deduce that the last integral in (22) can be written as

$$2\pi {\int }_{-\alpha }^{\alpha }{\left|\sum _{\left|\eta \right|\le T_v}c\left(\eta \right)e\left(\eta t\right)\right|}^{2}dt,$$

(23)

where $\alpha =(\gamma +log{x}_{v+1}-log{x}_v){\left(4\pi \right)}^{-1}$ and

$$c\left(\eta \right)=\left\{\begin{array}{cc}{e}^{i\eta \left(\gamma +log{x}_{v+1}+log{x}_v\right)/2},\hfill & \mathrm{if}\left|\eta \right|\le T_v;\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Applying Lemma 1 with $\Theta =1/2$ and $\delta =\Theta /\alpha \sim 2\pi /\gamma$ to estimate the integral in (23), we have

$$\begin{array}{cc}\hfill & \ll {\int }_{-\infty }^{\infty }{\left|{\delta }^{-1}\sum _{\genfrac{}{}{0pt}{}{t\le \eta \le t+\delta }{\left|\eta \right|\le T_v}}c\left(\eta \right)\right|}^{2}dt\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }{\delta }^{-2}\sum _{\genfrac{}{}{0pt}{}{t\le {\eta }_1,{\eta }_2\le t+\delta }{|{\eta }_1|,|{\eta }_2|\le T_v}}{e}^{i({\eta }_1-{\eta }_2)\left(\gamma +log{x}_{v+1}+log{x}_v\right)/2}dt\hfill \\ \hfill & \ll {\delta }^{-2}·{\delta }^2·T_v\ll x_v{log}^{-2A-3}{x}_v.\hfill \end{array}$$

(24)

From (20), (22) and (24), we get

$$|\Delta {\mathrm{\Psi }}_{2,c}\left(x_v\right)|\ll x_v{log}^{-A-2}{x}_v+x_v^c{log}^{2A+4}{x}_v.$$

This combined with (16) and (17) gives

$$E_1\ll x{log}^{-A-2}x.$$

(25)

Theorem 2 follows from (11), (12) and (25).

4. Proof of Theorem 1

To prove Theorem 1, we need Lemma 6 and Lemma 7. Furthermore, we only need to estimate $\mathrm{\Phi }(\gamma ,x_v,T_v)$ unconditionally. Note that

$$\mathrm{\Phi }(\gamma ,x_v,T_v)\ll log{x}_v·\underset{1\ll T\le T_v}{max}\left|{\mathrm{\Phi }}_1(\gamma ,x_v,T)\right|,$$

where

$${\mathrm{\Phi }}_1(\gamma ,x_v,T)=\sum _{x_v^c<m\le x_{v+1}^c}\mathrm{\Lambda }\left(m\right)\sum _{\genfrac{}{}{0pt}{}{\rho =\sigma +i\eta }{T/2<\left|\eta \right|\le T}}\frac{{(m+1)}^{\rho \gamma }-m^{\rho \gamma }}{\rho }.$$

We consider the following integral mean value of ${\mathrm{\Phi }}_1(\gamma ,x_v,T)$,

$$D({\gamma }_0,x_v,T)={\int }_{{\gamma }_0}^{{\gamma }_0+{log}^{-2}{x}_v}{\left|{\mathrm{\Phi }}_1(\gamma ,x_v,T)\right|}^{2}d\gamma$$

(26)

with ${\gamma }_0\in \left[\frac{1}{(\sqrt{5}+3)/8-\epsilon },\frac{1}{\epsilon }\right]$.

From Lemma 6, we know that the upper bound of S depends on the range of T, so we have the following three cases.

Case 1. When $T\ge x_v^{4/5}$.

In this case, we have $x_v^{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}<T$ and use (9) to get

$$\begin{array}{cc}\hfill D({\gamma }_0,x_v,T_v)\ll & x_v^{2/{\gamma }_0}{T}^{-3}{\left(x_v^{1/2}{T}^{5/4+\epsilon }\right)}^2=x_v^{1+2/{\gamma }_0}{T}^{-1/2+2\epsilon }\hfill \\ \hfill \ll & x_v^{1+3/2{\gamma }_0+2\epsilon /{\gamma }_0+(1/2-2\epsilon ){\delta }_1\left(\epsilon \right)}.\hfill \end{array}$$

Note that if $1/{\gamma }_0\le \left(\sqrt{5}+3\right)/8-\epsilon ,$ we have

$$\begin{array}{c}\hfill D({\gamma }_0,x_v,T)\ll x_v^{2-\epsilon }.\end{array}$$

Case 2. When $x_v^{9/20}\le T<x_v^{4/5}$.

In case of $x_v^{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}<T$, we use (9) and get

$$\begin{array}{cc}\hfill D({\gamma }_0,x_v,T_v)& \ll x_v^{2/{\gamma }_0}{T}^{-3}{\left(x_v^{3/2}{T}^{5/2+\epsilon }{e}^{-\sqrt{5log{x}_{v}logT}}\right)}^2\hfill \\ \hfill & =x_v^{3+2/{\gamma }_0}{T}^{2+2\epsilon }{e}^{-2\sqrt{5log{x}_{v}logT}}\hfill \\ \hfill & =e^{\left[(2+2\epsilon ){\left(\sqrt{\frac{logT}{log{x}_v}}\right)}^2-2\sqrt{5\frac{logT}{log{x}_v}}+3+\frac{2}{{\gamma }_0}\right]log{x}_v}\hfill \\ \hfill & =x_v^{(2+2\epsilon ){\left(\sqrt{\frac{logT}{log{x}_v}}\right)}^2-2\sqrt{5\frac{logT}{log{x}_v}}+3+2/{\gamma }_0},\hfill \end{array}$$

For convenience, we take $t=\sqrt{\frac{logT}{log{x}_v}}$ and consider a quadratic function $g\left(t\right)=(2+2\epsilon )t^2-2\sqrt{5}t+3+2/{\gamma }_0$.

If $x_v^{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}\le x_v^{9/20}\le T$, we have $t\in \left[\sqrt{9/20},\sqrt{4/5}\right)$ and $g\left(t\right)$ is monotonically decreasing in this interval. Hence,

$$g\left(t\right)\le g\left(\sqrt{9/20}\right)=9(1+\epsilon )/10+2/{\gamma }_0\le 9/5+9/10\epsilon +2{\delta }_1\left(\epsilon \right)$$

with $1/{\gamma }_0\le 9/20+{\delta }_1\left(\epsilon \right)$. Therefore,

$$D({\gamma }_0,x_v,T)\ll x_v^{2-\epsilon }.$$

If $x_v^{9/20}<x_v^{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}<T<x_v^{4/5}$, we get analogously $t\in \left(\sqrt{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)},\sqrt{4/5}\right)$ and

$$\begin{array}{cc}\hfill g\left(t\right)& <g\left(\sqrt{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}\right)=3+4/{\gamma }_0-2\sqrt{5\left(1/{\gamma }_0-{\delta }_1\left(\epsilon \right)\right)}+2\epsilon /{\gamma }_0-(2+2\epsilon ){\delta }_1\left(\epsilon \right)\hfill \\ \hfill & \le 3+4/{\gamma }_0-2\sqrt{5}/\sqrt{{\gamma }_0}+2\epsilon /{\gamma }_0+\left(\sqrt{10{\gamma }_0}-2-2\epsilon \right){\delta }_1\left(\epsilon \right)\hfill \\ \hfill & \le 2-\epsilon ,\hfill \end{array}$$

where we choose ${\delta }_1\left(\epsilon \right)$ sufficiently small and use the elementary inequality

$$\sqrt{1-x}\ge 1-x/\sqrt{2},\mathrm{for}x\in [0,1/2].$$

Therefore,

$$D({\gamma }_0,x_v,T)\ll x_v^{2-\epsilon }.$$

In case of $T\le x_v^{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}$, we use (10) and get

$$\begin{array}{cc}\hfill D({\gamma }_0,x_v,T)& \ll x_v^{3+2/{\gamma }_0}{T}^{2+2\epsilon }{e}^{-2\sqrt{5logTlog{x}_v}}{\left(T{x}_v^{-1/{\gamma }_0}\right)}^2\hfill \\ \hfill & =x_v^3{T}^{4+2\epsilon }{e}^{-2\sqrt{5logTlog{x}_v}}\hfill \\ \hfill & =x_v^{(4+2\epsilon ){\left(\sqrt{\frac{logT}{log{x}_v}}\right)}^2-2\sqrt{5\frac{logT}{log{x}_v}}+3},\hfill \end{array}$$

so we consider a new quadratic function $g_1\left(t\right)=(4+2\epsilon )t^2-2\sqrt{5}t+3$.

If $x_v^{9/20}\le T\le x_v^{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}<x_v^{4/5}$, we get $t\in \left[\sqrt{9/20},\sqrt{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}\right]$ and

$$\begin{array}{cc}\hfill g_1\left(t\right)& \le g_1\left(1/{\gamma }_0-{\delta }_1\left(\epsilon \right)\right)=4\left(1/{\gamma }_0-{\delta }_1\left(\epsilon \right)\right)-2\sqrt{5}·\sqrt{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}+3+2\epsilon /{\gamma }_0-2\epsilon {\delta }_1\left(\epsilon \right)\hfill \\ \hfill & \le 4/{\gamma }_0-2\sqrt{5}/\sqrt{{\gamma }_0}+3+2\epsilon /{\gamma }_0+\left(\sqrt{10{\gamma }_0}-4-2\epsilon \right){\delta }_1\left(\epsilon \right)\hfill \\ \hfill & \le 2-\epsilon .\hfill \end{array}$$

Therefore,

$$D({\gamma }_0,x_v,T)\ll x_v^{2-\epsilon }.$$

Case 3. When $T<x_v^{9/20}$.

If $x_v^{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}<T<x_v^{9/20}$, we use (9) to get

$$x_v^{2/{\gamma }_0}{T}^{-3}{\left(x_v^{3/4}{T}^{5/6+\epsilon }\right)}^2=x_v^{3/2+2/{\gamma }_0}{T}^{-4/3+2\epsilon }\ll x_v^{2-\epsilon }$$

and

$$x_v^{2/{\gamma }_0}{T}^{-3}{\left(x_v^{1-\beta }{T}^{3\beta +\epsilon }\right)}^2=x_v^{2-2\beta +2/{\gamma }_0}{T}^{-3+6\beta +2\epsilon }\ll x_v^{2-\epsilon }.$$

Therefore,

$$D({\gamma }_0,x_v,T)\ll x_v^{2/{\gamma }_0}{T}^{-3}{\left(x_v^{3/4}{T}^{5/6+\epsilon }+x_v^{1-\beta }{T}^{3\beta +\epsilon }\right)}^2\ll x_v^{2-\epsilon }.$$

If $T\le x_v^{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}<x_v^{9/20}$ or $T<x_v^{9/20}\le x_v^{1/{\gamma }_0-{\delta }_1\left(\epsilon \right)}$, we use (10) to get

$$x_v^{2/{\gamma }_0}{T}^{-3}{\left(x_v^{3/4}{T}^{5/6+\epsilon }\right)}^2{\left(T{x}_v^{-1/{\gamma }_0}\right)}^2\ll x_v^{3/2}{T}^{2/3+2\epsilon }\ll x_v^{2-\epsilon }$$

and

$$x_v^{2/{\gamma }_0}{T}^{-3}{\left(x_v^{1-\beta }{T}^{3\beta +\epsilon }\right)}^2{\left(T{x}_v^{-1/{\gamma }_0}\right)}^2\ll x_v^{2-2\beta }{T}^{-1+6\beta +2\epsilon }\ll x_v^{2-\epsilon }.$$

Therefore,

$$D({\gamma }_0,x_v,T)\ll x_v^{2/{\gamma }_0}{T}^{-3}{\left(x_v^{3/4}{T}^{5/6+\epsilon }+x_v^{1-\beta }{T}^{3\beta +\epsilon }\right)}^2{\left(T{x}_v^{-1/{\gamma }_0}\right)}^2\ll x_v^{2-\epsilon }.$$

Combining all the above cases, we obtain

$$D({\gamma }_0,x_v,T)={\int }_{{\gamma }_0}^{{\gamma }_0+{log}^{-2}{x}_v}{\left|{\mathrm{\Phi }}_1(\gamma ,x_v,T)\right|}^{2}d\gamma \ll x_v^{2-\epsilon }.$$

Inequality (20) gives us

$$\begin{array}{cc}\hfill {\int }_{{\gamma }_0}^{{\gamma }_0+{log}^{-2}{x}_v}{\left|{\mathrm{\Psi }}_{2,c}\left(x\right)\right|}^{2}d\gamma \ll & {log}^{4}x·{\int }_{{\gamma }_0}^{{\gamma }_0+{log}^{-2}{x}_v}{|\Delta {\mathrm{\Psi }}_{2,c}\left(x_v\right)|}^{2}d\gamma \hfill \\ \hfill \ll & {log}^{4}x·{\int }_{{\gamma }_0}^{{\gamma }_0+{log}^{-2}{x}_v}{\left|\mathrm{\Phi }(\gamma ,x_v,T_v)\right|}^{2}d\gamma +x_v^{2c}{log}^{4A+10}x\hfill \\ \hfill \ll & {log}^{6}x·\underset{1\ll T\le T_v}{max}{\int }_{{\gamma }_0}^{{\gamma }_0+{log}^{-2}{x}_v}{\left|{\mathrm{\Phi }}_1(\gamma ,x_v,T)\right|}^{2}d\gamma +x_v^{2c}{log}^{4A+10}x\hfill \\ \hfill \ll & x^{2-\epsilon }.\hfill \end{array}$$

Let ${\gamma }_0:=\frac{1}{(\sqrt{5}+3)/8-\epsilon },{\gamma }_{i+1}:={\gamma }_i+\frac{1}{{log}^2{x}_v}$ and ${\mathcal{A}}_i:=\{\gamma \in [{\gamma }_i,{\gamma }_{i+1}]/|{\mathrm{\Psi }}_{2,c}\left(x\right)|>x{log}^{-A}x\}$. Then by Tschebytschev’s inequality, we obtain

$$\begin{array}{cc}\hfill \mu \left({\mathcal{A}}_i\right)& \ll x^{-2}{log}^{2A}x·{\int }_{{\gamma }_i}^{{\gamma }_{i+1}}{\left|{\mathrm{\Psi }}_{2,c}\left(x\right)\right|}^{2}d\gamma \hfill \\ \hfill & \ll x^{-\epsilon }{log}^{2A}x\ll x^{-\epsilon }.\hfill \end{array}$$