\( \def\wtilde{\widetilde} \def\cat{\textrm} \def\proved{} \def\lbrackk{[\![} \def\rbrackk{]\!]} \def\qandq{\quad \text{and} \quad}\newcommand{\cvdots}[1][]{\quad\ \vdots} \newcommand{\mfrac}{\frac} \newcommand{\dsty}{\displaystyle} \newcommand{\ipfrac}[3][]{\tfrac{1}{#3}(#2)} \newcommand{\unfrac}[2]{{#1}/{#2}} \newcommand{\punfrac}[2]{\left({#1}/{#2}\right)} \newcommand{\upnfrac}[2]{{(#1)}/{#2}} \newcommand{\unpfrac}[2]{{#1}/{(#2)}} \newcommand{\upnpfrac}[2]{{(#1)}/{(#2)}} \newcommand{\parbox}[2]{\text{#2}} \newcommand{\ssty}[1]{{\scriptstyle{#1}}} \def\liso{\ \tilde{\longrightarrow}\ } \def\tsum{\sum} \def\mbinom{\binom} \) \( \def\C{\mathbb C} \def\R{\mathbb R} \def\Z{\mathbb Z} \def\Q{\mathbb Q} \def\F{\mathbb F} \def\smash{} \def\phantomplus{} \def\nobreak{} \def\omit{} \def\hidewidth{} \renewcommand{\mathnormal}{} \renewcommand{\qedhere}{} \def\sp{^} \def\sb{_} \def\vrule{|} \def\hrule{} \def\dag{\dagger} \def\llbracket{[\![} \def\rrbracket{]\!]} \def\llangle{\langle\!\langle} \def\rrangle{\rangle\!\rangle} \def\sssize{\scriptsize} \def\mathpalette{} \def\mathclap{} \def\coloneqq{\,:=\,} \def\eqqcolon{\,=:\,} \def\colonequals{\,:=\,} \def\equalscolon{\,=:\,} \def\textup{\mbox} \def\makebox{\mbox} \def\vbox{\mbox} \def\hbox{\mbox} \def\mathbbm{\mathbb} \def\bm{\boldsymbol} \def\/{} \def\rq{'} \def\lq{`} \def\noalign{} \def\iddots{\vdots} \def\varint{\int} \def\l{l} \def\lefteqn{} \def\slash{/} \def\boxslash{\boxminus} \def\ensuremath{} \def\hfil{} \def\hfill{} \def\dasharrow{\dashrightarrow} \def\eqno{\hskip 50pt} \def\curly{\mathcal} \def\EuScript{\mathcal} \def\widebar{\overline} \newcommand{\Eins}{\mathbb{1}} \newcommand{\textcolor}[2]{#2} \newcommand{\textsc}[1]{#1} \newcommand{\textmd}[1]{#1} \newcommand{\emph}{\text} \newcommand{\uppercase}[1]{#1} \newcommand{\Sha}{{III}} \renewcommand{\setlength}[2]{} \newcommand{\raisebox}[2]{#2} \newcommand{\scalebox}[2]{\text{#2}} \newcommand{\stepcounter}[1]{} \newcommand{\vspace}[1]{} \newcommand{\displaybreak}[1]{} \newcommand{\textsl}[1]{#1} \newcommand{\prescript}[3]{{}^{#1}_{#2}#3} \def\llparenthesis{(\!\!|} \def\rrparenthesis{|\!\!)} \def\ae{a\!e} \def\nolinebreak{} \def\allowbreak{} \def\relax{} \def\newline{} \def\iffalse{} \def\fi{} \def\func{} \def\limfunc{} \def\mathbold{\mathbf} \def\mathscr{\mathit} \def\bold{\mathbf} \def\dvtx{\,:\,} \def\widecheck{\check} \def\spcheck{^\vee} \def\sphat{^{{}^\wedge}} \def\degree{{}^{\circ}} \def\tr{tr} \def\defeq{\ :=\ } \newcommand\rule[3][]{} \newcommand{\up}[1]{\textsuperscript{#1}} \newcommand{\textsuperscript}[1]{^{#1}} \newcommand{\fracwithdelims}[4]{\left#1\frac{#3}{#4}\right#2} \newcommand{\nicefrac}[2]{\left. #1\right/#2} \newcommand{\sfrac}[2]{\left. #1\right/#2} \newcommand{\discretionary}[3]{#3} \newcommand{\xlongrightarrow}[1]{\xrightarrow{\quad #1\quad}} \def\twoheadlongrightarrow{ \quad \longrightarrow \!\!\!\!\to \quad } \def\xmapsto{\xrightarrow} \def\hooklongrightarrow{\ \quad \hookrightarrow \quad \ } \def\longlonglongrightarrow{\ \quad \quad \quad \longrightarrow \quad \quad \quad \ } \def\rto{ \longrightarrow } \def\tto{ \longleftarrow } \def\rcofib{ \hookrightarrow } \def\L{\unicode{x141}} \def\niplus{\ \unicode{x2A2E}\ } \def\shuffle{\ \unicode{x29E2}\ } \def\fint{{\LARGE \unicode{x2A0F}}} \def\XXint#1#2#3{\vcenter{\hbox{$#2#3$}}\kern-0.4cm} \newcommand{\ve}{\varepsilon} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\R}{\mathbb R} \newcommand{\Z}{\mathbb Z} \newcommand{\Q}{\mathbb Q} \renewcommand{\H}{\mathcal H} \newcommand{\Pabn}{P_n^{(a,b)}} \def\logg{\log_{(2)}} \def\loggg{\log_{(3)}} \newcommand{\mm}[4]{\begin{pmatrix} #1 & #2 \cr #3 & #4 \end{pmatrix}} \newcommand{\ontop}[2]{\genfrac{}{}{0pt}{}{#1}{#2}} \)

Section 3Proofs of technical results

In this section we provide the proofs of Proposition 2.1 and Proposition 2.1.2.

Subsection 3.1Proof of Proposition 2.1.2

A special case of the Proposition is the following:

Claim3.1.1

There exists \(C_1>0\) such that the number of \(\gamma_n< T\) satisfying

\begin{equation}N\left(\gamma_n+\frac{lC^*\logg T}{\log T}\right)-N\left(\gamma_n+\frac{(lC^*-1)\logg T}{\log T}\right)\le C_1\logg T,\quad\quad\tag{3.1.1} \end{equation}

for all \(|l|\le\log T/(C^*\logg T)\), is

\begin{equation}\frac{T}{2\pi}\log T+O\left(\frac{T}{(\log T)^{m_0}}\right).\quad\quad\tag{3.1.2} \end{equation}

Here \(C_1\) is not depending on \(C^*\).

The proof of Claim follows easily from the same method below. Thus, we omit the proof of it.

From now on, we are assuming that \(\gamma_n\) satisfies \(T/(\log T)^{m_0+1}< \gamma_n< T\) and Claim. We recall

\begin{align}\int_T^{T+H}|S(t+h)-S(t)|^{2k}dt=\mathstrut&\frac{H(2k)!}{(2\pi^2)^kk!}\log^k(2+h\log T)\quad\quad\quad\quad\quad\quad \quad \tag{3.1.3}\\ &+O\left(H(ck)^k\left(k+\log^{k-1/2}(2+h\log T)\right)\right)\quad\quad\quad\quad\quad\quad \quad \tag{3.1.4} \end{align}

uniformly for \(T^a< H\le T\), \(a>1/2\), \(0< h< 1\) and any positive integer \(k\), where \(c\) is a positive constant and \(S(t)=\frac{1}{\pi}\arg\zeta(1/2+it)\). For this, see [16, Theorem 4]. Thus we have

\begin{equation} \int_0^T|S(t+h)-S(t)|^{2k}dt\ll T\left(Ak\right)^{2k},\quad\quad\tag{3.1.5} \end{equation}

where \(\log(2+h\log T)\ll k\). We note that

\begin{equation} S(t+h)-S(t)=N(t+h)-N(t)-\frac{h}{2\pi}\log t+O\left(\frac{h^2+1}{t}\right),\quad\quad\tag{3.1.6} \end{equation}

where \(N(t)\) is the number of zeros of \(\zeta(s)\) in \(0< \Im s< t\). By this, we have

\begin{equation}\widetilde{S}(t,l_1,l_2)=N\left(t+\frac{(l_2-l_1)C^*\logg T}{\log T}\right)-N(t)-\frac{(l_2-l_1)C^*\logg T}{2\pi}+O\left(\frac{1}{t}\right), \end{equation}

where

\begin{equation}\widetilde{S}(t,l_1,l_2)=S\left(t+\frac{(l_2-l_1)C^*\logg T}{\log T}\right)-S(t)\quad\quad\tag{3.1.7} \end{equation}

Using Claim, the last formula and (3.1.6), we have

\begin{align}N(n,l_1,l_2)\le\mathstrut &\left|\widetilde{S}(t,l_1,l_2)\right|+\logg T\quad\quad\quad\quad\quad\quad \quad \tag{3.1.8}\\ \le\mathstrut &\left|\widetilde{S}(t-h,l_1,l_2)\right|+3\logg T\quad\quad\quad\quad\quad\quad \quad \tag{3.1.9}\\ &\mathstrut +\sum_{j=1}^{2}N\left(\gamma_n+\frac{l_jC^*\logg T}{\log T}\right)-N\left(\gamma_n+\frac{(l_jC^*-1)\logg T}{\log T}\right)\quad\quad\quad\quad\quad\quad \quad \tag{3.1.10}\\ \le\mathstrut &C_2\logg T+\left|\widetilde{S}(t-h,l_1,l_2)\right|\quad\quad\quad\quad\quad\quad \quad \tag{3.1.11} \end{align}

for \(t=\gamma_n+l_1C^*\logg T/\log T\) and \(0\le h\le\logg T/\log T\), where \(C_2=\max\{2C_1+3,A\}\). Using this, we have

\begin{align}\sum_{\substack{\frac{T}{(\log T)^{m_0+1}}< \gamma_n< T\\ N(n,l_1,l_2)\geqslant C\logg T} }(C\logg T)^{2k} \mathstrut & \mathstrut \frac{\logg T}{\log T}\quad\quad\quad\quad\quad\quad \quad \tag{3.1.12}\\ \ll \mathstrut &T\log T(2C_2\logg T)^{2k}+\sum_{\gamma_n< T}\int_{\gamma_n+\frac{(l_1C^*-1)\logg T}{\log T}}^{\gamma_n+\frac{l_1C^*\logg T}{\log T}}\left|2\widetilde{S}(t,l_1,l_2)\right|^{2k}dt\quad\quad\quad\quad\quad\quad \quad \tag{3.1.13}\\ \ll \mathstrut & T\log T(2C_2\logg T)^{2k} +\log T\int_0^T\left|2\widetilde{S}(t,l_1,l_2)\right|^{2k}dt\quad\quad\quad\quad\quad\quad \quad \tag{3.1.14}\\ \ll \mathstrut &T\log T\left((2C_2\logg T)^{2k}+(2C_2k)^{2k}\right)\quad\quad\quad\quad\quad\quad \quad \tag{3.1.15} \end{align}

for any sufficiently large \(T\) and any \(|l_1|,|l_2|\le\log T/(C^*\logg T)\) with \(0< l_2-l_1\le 2\log T/(C^*\logg T)\). We put

\begin{equation}k=[\logg T]\qquad\text{and} \qquad C=e^{m_0+2}(2C_2+1).\quad\quad\tag{3.1.16} \end{equation}

By these and the last inequality, we have

\begin{equation}\sum_{\substack{|l_1|,|l_2|\le\frac{\log T}{C^*\logg T}\\0< l_2-l_1\le\frac{2\log T}{C^*\logg T}} } \sum_{\substack{\frac{T}{(\log T)^{m_0+1}}< \gamma_n< T\\ N(n,l_1,l_2)\geqslant C\logg T} }1\ll\frac{T(\log T)^4(2C_2\logg T)^{2k}}{\left(C\logg T\right)^{2k}}\ll\frac{T}{(\log T)^{m_0}}. \end{equation}

We complete the proof of Proposition 2.1.2.

Subsection 3.2Proof of Proposition 2.1

We recall

\begin{equation}\frac{\zeta'}{\zeta}(s)=O(\log t)+\sum_{|\gamma-t|\le1}\frac{1}{s-\rho}\quad\quad\tag{3.2.1} \end{equation}

holds uniformly for \(t>1\) and \(-2\le\R s\le1\). For this, see [15, Theorem 9.6 (A)]. Using the last formula, it suffices to show that the number of \(\gamma_n\) in \(T/(\log T)^{m_0+1}\le\gamma_n< T\) such that \(\gamma_n\) satisfies the condition in Proposition 2.1.2 and

\begin{equation}\sum_{\frac{C^*\logg T}{\log T}< |\gamma_n-\gamma_m|\le1}\frac{1}{\gamma_n-\gamma_m}=O(\log T)\quad\quad\tag{3.2.2} \end{equation}

is

\begin{equation}\frac{T}{2\pi}\log T+O\left(\frac{T}{(\log T)^{m_0}}\right)\qquad(T\to\infty),\quad\quad\tag{3.2.3} \end{equation}

because for \(s=1/2+1/\log T+it\) and \(|\gamma_n-t|\le A/\log T\), we have

\begin{equation}\sum_{\frac{C^*\logg T}{\log T}< |\gamma_n-\gamma_m|\le1}\frac{1}{s-\rho}-\frac{1}{i(\gamma_n-\gamma_m)}=O\left(\frac{1}{\log T}\sum_{m=1}^{\infty}\frac{\logg T}{\left(\frac{m\logg T}{\log T}\right)^2}\right)=O(\log T). \end{equation}

We recall that Proposition 2.1.2 implies

\begin{equation}N\left(\gamma_n+\frac{lC^*\logg T}{\log T}\right) =N(\gamma_n)+\frac{lC^*\logg T}{2\pi}+O\left(\logg T\right)\quad\quad\tag{3.2.4} \end{equation}

for any integer \(l\) with \(|l|\le\log T/(C^*\logg T)\). This immediately implies that for a sufficiently large \(C^*>1\), we have

\begin{equation}\max_{0\le k\le N}|2\gamma_n-\gamma_{m_2+k}-\gamma_{m_1-k}|\frac{\log T}{2\pi}=O(\logg T),\quad\quad\tag{3.2.5} \end{equation}

where \(\gamma_{m_1}\) is the greatest one in \([\gamma_n-1,\gamma_n-C^*\logg T/\log T)\), \(\gamma_{m_1}\) the least one in \((\gamma_n+C^*\logg T/\log T,\gamma_n+1]\) and \(N\) the largest positive integer such that \(\gamma_{m_1-N}\) and \(\gamma_{m_2+N}\) belong to \([\gamma_n-C^*\logg T/\log T,\gamma_n+C^*\logg T/\log T]\). By this and putting \( M(n)=\max_{0\le k\le N}|2\gamma_n-\gamma_{m_2+k}-\gamma_{m_1-k}|,\) we have

\begin{equation}M(n)\ll\frac{\logg T}{\log T}\quad\quad\tag{3.2.6} \end{equation}

Using this and Proposition 2.1.2 and the fact [15, Theorems 9.3 and 14.13] that the number of zeros between \(t\) and \(t+1\) is

\begin{equation}\frac{\log t}{2\pi}+O\left(\frac{\log t}{\logg t}\right)\qquad(\text{as } t\to\infty),\quad\quad\tag{3.2.7} \end{equation}

we have

\begin{align}\sum_{\frac{C^*\logg T}{\log T}< |\gamma_n-\gamma_m|\le1}\frac{1}{\gamma_n-\gamma_m}=\mathstrut&\sum_{0\le k\le N}\frac{2\gamma_n-\gamma_{m_2+k}-\gamma_{m_1-k}}{(\gamma_n-\gamma_{m_2+k})(\gamma_n-\gamma_{m_1-k})}+O(\log T)\quad\quad\quad\quad\quad\quad \quad \tag{3.2.8}\\ =\mathstrut&O\left(M(n)\sum_{k=1}^{\infty}\frac{\logg T}{\left(\frac{k\logg T}{\log T}\right)^2}\right)+O(\log T)\quad\quad\quad\quad\quad\quad \quad \tag{3.2.9}\\ =\mathstrut&O(\log T).\quad\quad\quad\quad\quad\quad \quad \tag{3.2.10} \end{align}

Thus, we complete the proof of Proposition 2.1.