Related papers: Experiments with the dynamics of the Riemann zeta …
Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$. Our proof is based upon a recent method of Rudnick and Soundararajan…
We show that the analytic continuations of Helson zeta functions $ \zeta_\chi (s)= \sum_1^{\infty}\chi(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole…
In this paper we investigate analytic inequalities related to a conjecture of Henry involving the difference between the Riemann zeta function and the digamma function. By treating $\zeta(s)-\psi(1-s)$ as a unified analytic object, we…
This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is…
In this work, we estimate the sum \begin{align*} \sum_{0 < \Im(\rho) \leq T} \zeta(\rho+\alpha)X(\rho) Y(1\!-\! \rho) \end{align*} over the nontirival zeros $\rho$ of the Riemann zeta funtion where $\alpha$ is a complex number with…
The paper reviews existing results about the statistical distribution of zeros for the three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of…
In this paper is stablished a characterization of the solutions of the equation: zeta(z) = 0. Then such a characterization is used to give a proof for Riemann is Conjecture.
Let $X$ be a smooth proper curve over a finite field and let $\infty \in X$ be a closed point. Let $A$ be the ring of functions on $X - \infty$. The Goss zeta function $\zeta_A$ of $A$ is an equicharacteristic analogue of the Riemann zeta…
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…
In this manuscript, we consider the Riemann zeta function $\zeta$, defined through the Abel summation formula. We present a simple analytical method based on a complex differential equation. The aim is to propose a new analytical approach,…
Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$ for every positive integer k. Our bounds are nearly as sharp as…
In these lectures we first review the important properties of the Riemann $\zeta$-function that are necessary to understand the nature and importance of the Riemann hypothesis (RH). In particular this first part describes the analytic…
The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in Random Matrix Theory and…
We study lower bounds for the Riemann zeta function $\zeta(s)$ along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the…
Finding hidden order within disorder is a common interest in material science, wave physics, and mathematics. The Riemann hypothesis, stating the locations of nontrivial zeros of the Riemann zeta function, tentatively characterizes…
We study the relationship between the zeros of the Riemann zeta function and physical systems exhibiting supersymmetry, $PT$ symmetry and $SU(2)$ group symmetry. Our findings demonstrate that unbroken supersymmetry is associated with the…
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.
A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending…
Recently, we have established the generalized Li criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_(/rho)(1-(1-((/rho-a)/(/rho+a-1))^n) for any real a not…
This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…