
## Section3Proofs of technical results

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

### Subsection3.1Proof 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

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

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

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

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

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

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

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

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

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

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

where

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

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

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

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

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

By these and the last inequality, we have

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

We complete the proof of Proposition 2.1.2.

### Subsection3.2Proof of Proposition 2.1

We recall

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

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

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

is

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

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

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

We recall that Proposition 2.1.2 implies

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

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

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

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

$$M(n)\ll\frac{\logg T}{\log T}\quad\quad\tag{3.2.6}$$

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

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

we have