Related papers: Large gaps between consecutive zeros of the Rieman…

In this paper, we study a more general pair correlation function, $F_h(x,T)$, of the zeros of the Riemann zeta function. It provides information on the distribution of larger differences between the zeros.

We show that the Generalized Riemann Hypothesis for all Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of $\zeta(s)$ satisfy those…

In this paper, a positive answer to the Riemann hypothesis is given by using a new result that predict the exact location of zeros of the alternating zeta function on the critical strip.

This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens' conjecture. We show that $\limsup M(x)x^{-1/2} \geq 1.6383$ and that $\liminf…

We establish an unconditional asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For $\Re(s)=\sigma$ satisfying $(\log T)^{-1/3+\epsilon} \leq (2\sigma-1) \leq (\log \log…

It is shown that, for every integer n>2, there exists \delta_{n}>0 such that the closure of the set of the real parts of the zeros of the nth partial sum of the Riemann zeta function \zeta_{n} contains to the interval [-\delta_{n},b^{n}].…

Using the twisted fourth moment of the Riemann zeta-function we study large gaps between consecutive zeros of the derivatives of Hardy's function $Z(t)$, improving upon previous results of Conrey and Ghosh [J. London Math. Soc. 32 (1985),…

By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.

The following theorem is proven: Both real and imaginary parts of the function F(s) defined as F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), and whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function,…

We show assuming RH that phenomena concerning pairs of zeros established $via$ pair correlations occur with positive density (with at most a slight adjustment of the constants). Also, while a double zero is commonly considered to be a close…

A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective…

We continue our investigation of the distribution of the fractional parts of $a \gamma$, where $a$ is a fixed non-zero real number and $\gamma$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We…

On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Soshnikov and others in…

The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented.

Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function $\zeta(s)$. They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line $\sigma…

Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments…

We consider the alternating Riemann zeta function $\zeta^*(s)= \sum^{\infty} _{ n=1} \frac{(-1)^{n-1}}{n^s}$, which converges if $Re (s)>0 .$ By using Rouche's theorem, the Bolzano-Weierstrass theorem and by method of contradiction we…

The Riemann Hypothesis is a conjecture made in 1859 by the great mathematician Riemann that all the complex zeros of the zeta function $\zeta(s)$ lie on the `critical line' ${Rl} s= 1/2$. Our analysis shows that the assumption of the truth…

Analyzing in detail the analytic continuation of the Riemann zeta function we are able to generate several new identities which may be useful for application in physics and mathematics.

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…