Related papers: Statistical regularities in the zeta zeros
We present a conjecture about the asymptotic representation of certain series. The conjecture implies the Riemann hypothesis and it would also indicate the simplicity of the non-trivial zeros of the zeta-function.
Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We…
In this paper, we will give a new proof for a known result of the mean square of Riemann zeta-function.
The alternating zeta function zeta*(s) = 1 - 2^{-s} + 3^{-s} - ... is related to the Riemann zeta function by the identity (1-2^{1-s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 1-2^{1-s}…
The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann zeta function which are sufficiently strong to break the density hypothesis in a nontrivial part of the critical strip. Apart from…
We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann Hypothesis for curves, the zeros all lie…
We settle a conjecture of Farmer and Ki in a stronger form. Roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta-function $\zeta(s)$ if and only if there is a positive proportion…
We consider the statistical distribution of zeros of random meromorphic functions whose poles are independent random variables. It is demonstrated that correlation functions of these zeros can be computed analytically and explicit…
The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta…
This paper is a summary of the general approach outlined in my previous papers toward proving the riemann hypothesis. Numerical and graphical proof of the Riemann Hypothesis is presented with analytical arguments although more work needs…
Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of $\zeta(s)$. Moreover, with additional assumptions, we prove the existence of a limiting…
In this paper, an elementary method to find the values of the Riemann Zeta function at even natural numbers, and to find values of a closely related series at odd natural numbers is presented. Another method, specifically for the evaluation…
The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of…
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.
We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz…
We introduce a new Tauberian framework through the theory of "regular arithmetic functions". This allows us to establish a characterization of the Riemann hypothesis by linking the floor function to the distribution of nontrivial zeros of…
This paper considers some infinite series involving the Riemann zeta function.
We develop a finite-dimensional, symmetric matrix framework associated with the Riemann zeta function for complex arguments s with Real(s) unequal 1/2.
Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin…
Some identities for the Riemann zeta-function are proved, using properties of the Mellin transform and M\"untz's identity.