Related papers: Linear statistics for zeros of Riemann's zeta func…

In this article, we count the number of consecutive zeros of the Epstein zeta-function, associated to a certain quadratic form, on the critical line with ordinates lying in $[0,T], T$ sufficiently large and which are separated apart by a…

Let $Z(t)$ be the classical Hardy function in the theory of the Riemann zeta-function. The main result in this paper is that if the Riemann hypothesis is true then for any positive integer $n$ there exists a $t_{n}>0$ such that for…

We calculate the discrete moments of the characteristic polynomial of a random unitary matrix, evaluated a small distance away from an eigenangle. Such results allow us to make conjectures about similar moments for the Riemann zeta…

The purpose of the present paper is to reveal the relation between the behavior of the logarithm of the Riemann zeta-function $\log{\zeta(s)}$ and the distribution of zeros of the Riemann zeta-function. We already know some examples for the…

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $\frac12+it$. Assuming the Riemann Hypothesis, we sharpen the constant in the best currently known bounds for $S(t)$ and for the change of $S(t)$ in intervals. We…

In 2007, assuming the Riemann Hypothesis (RH), Soundararajan \cite{Moment} proved that $\int_{0}^T |\zeta(1/2 + it)|^{2k} dt \ll_{k, \epsilon} T(\log T)^{k^2 + \epsilon}$ for every $k$ positive real number and every $\epsilon > 0.$ In this…

We present a quantum mechanical model which establishes the veracity of the Riemann hypothesis that the non-trivial zeros of the Riemann zeta-function lie on the critical line of $\zeta(s)$.

In the present paper the asymptotic formulae for the first moment of the Riemann zeta-function on the critical line is proven under assumption of the Riemann Hypothesis.

A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structure, plus countably many special…

We study a problem of estimation of smooth functionals of parameter $\theta $ of Gaussian shift model $$ X=\theta +\xi,\ \theta \in E, $$ where $E$ is a separable Banach space and $X$ is an observation of unknown vector $\theta$ in Gaussian…

In this work, we present a non-linear difference equation for calculation of the zeros of the Riemann's zeta-function on the critical line. Our proposed non-linear map uses the Lambert W function and it can be easily implemented in a…

Here, we study both analytically and numerically, an integral $Z(\sigma,r)$ related to the mean value of a generalized moment of Riemann's zeta function. Analytically, we predict finite, but discontinuous values and verify the prediction…

In this paper, we introduce a geometrical summation method that makes the original Riemann series converge over the critical strip. This method gives an analytical function, that coincides with z\^eta. This point of view allows us to…

Assuming the Riemann hypothesis we demonstrate the existence of smooth numbers in certain short intervals.

We numerically investigate, for zeros $\rho=1/2+i\gamma$, the statistics of the imaginary part of $\log(\zeta^\prime(1/2+i\gamma))$, computed by continuous variation along a vertical line from $\sigma=4$ to $4+i\gamma$ and then along a…

We investigate a dynamical basis for the Riemann hypothesis (RH) that the non-trivial zeros of the Riemann zeta function lie on the critical line x = 1/2. In the process we graphically explore, in as rich a way as possible, the diversity of…

We present drawings on the complex plane of the lines Im(zeta(s))=0 and Re(zeta(s))=0. This allow to illustrate many properties of the zeta function of Riemann. This is an expository paper. It does not pretend to prove any new result about…

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.

In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of…

In this paper, we focus on the existence of accumulation points of the subset defined by the real projection of the zeros of the partial sums of the Riemann zeta functions. That would imply the existence of an infinite amount of zeros of…