相关论文: On the number of zeros of $\mathop{\mathcal R}(s)$

In a letter to Weierstrass Riemann asserted that the number $N_0(T)$ of zeros of $\zeta(s)$ on the critical line to height $T$ is approximately equal to the total number of zeros to this height $N(T)$. Siegel studied some posthumous papers…

We state and give complete proof of the results of Siegel about the zeros of the auxiliary function of Riemann $\mathop{\mathcal R}(s)$. We point out the importance of the determination of the limit to the left of the zeros of…

Assuming the Riemann Hypothesis, we prove that $$ N_1(T) = \frac{T}{2\pi}\log \frac{T}{4\pi e} + O\bigg(\frac{\log T}{\log\log T}\bigg), $$ where $N_1(T)$ is the number of zeros of $\zeta'(s)$ in the region $0<\Im s\le T$.

We prove a density theorem for the auxiliar function $\mathop{\mathcal R}(s)$ found by Siegel in Riemann papers. Let $\alpha$ be a real number with $\frac12< \alpha\le 1$, and let $N(\alpha,T)$ be the number of zeros $\rho=\beta+i\gamma$ of…

We have computed all zeros $\beta+i\gamma$ of $\mathop{\mathcal R }(s)$ with $0<\gamma<215946.3$. A total of 162215 zeros with 25 correct decimal digits. In this paper we offer some statistic based on this set of zeros. Perhaps the main…

We show that there is a sequence of zeros of $\mathop{\mathcal R}(s)$ in the fourth quadrant. We show that the $n$-th zero $\rho_{-n}=\beta_{-n}+i\gamma_{-n}$, with $\beta_{-n}\sim 4\pi^2 n/\log^2n$ and $\gamma_{-n}\sim-4\pi n/\log n$. We…

Assuming the Riemann hypothesis, we prove that $$ N_k(T) = \frac{T}{2\pi}\log \frac{T}{4\pi e} + O_k\left(\frac{\log{T}}{\log\log{T}}\right), $$ where $N_k(T)$ is the number of zeros of $\zeta^{(k)}(s)$ in the region $0<\Im s\le T$. We…

The research shows that Riemann proved that all of zeros of Riemann's zeta function are on $\sigma=1/2$ based on the functional equation \begin{align*} \pi^{-\frac{s}{2}}\Gamma \left( \frac{s}{2} \right) \zeta(s)&={\frac{1}{s(s-1)} +…

In this article, we show that $$ \left| N (T) - \frac{T}{ 2 \pi} \log \left( \frac{T}{2\pi e}\right) \right| \le 0.1038 \log T + 0.2573 \log\log T + 9.3675 $$ where $N(T)$ denotes the number of non-trivial zeros $\rho$, with $0<\Im(\rho)…

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log…

In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing \[ \left| N (T) - \frac{T}{ 2 \pi} \log \left( \frac{T}{2\pi e}\right) \right|\le 0.10076\log T+0.24460\log\log T+8.08344, \] for every $T\ge e$,…

There are many analytic functions $U(t)$ satisfying $Z(t)=2\Re\bigl\{ e^{i\vartheta(t)}U(t)\bigr\}$. Here, we consider an entire function $\mathop{\mathcal L}(s)$ such that $U(t)=\mathop{\mathcal L}(\frac12+it)$ is one of the simplest among…

We study the horizontal distribution of zeros of $\zeta'(s)$ which are denoted as $\rho'=\beta'+i\gamma'$. We assume the Riemann hypothesis which implies $\beta'\geqslant1/2$ for any non-real zero $\rho'$, equality being possible only at a…

It is proved that $s=-2n$ is a simple zero of $\mathop{\mathcal R}(s)$ for each integer $n\ge1$. Here $\mathop{\mathcal R}(s)$ is the function found by Siegel in Riemann's posthumous papers.

Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…

Let $0<\gamma_1\leq \gamma_2\leq \ldots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $\Theta, \vartheta>0$ such that for…

Let $\mathop{\mathcal R}(s)$ be the function related to $\zeta(s)$ found by Siegel in the papers of Riemann. In this paper we obtain the main terms of the mean values \[\frac{1}{T}\int_0^T |\mathop{\mathcal…

Assume the Riemann Hypothesis, and let $\gamma^+>\gamma>0$ be ordinates of two consecutive zeros of $\zeta(s)$. It is shown that if $\gamma^+-\gamma < v/ \log \gamma $ with $v<c$ for some absolute positive constant $c$, then the box $$…

Let $y\ne 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_*>0$ $($depending on $y$ and $C)$ such that for every $T\ge T_*$, both \[ \zeta(\tfrac12+i\gamma)=0 \quad\text{and}\quad\zeta(\tfrac12+i(\gamma+y))\ne 0 \] hold for…

A discussion involving the evaluation of the sum $$\sum_{T<\g\le T+H}|\zeta(1/2+i\gamma)|^2$$ and some related integrals is presented, where $\gamma\,(>0)$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. It…