Related papers: Experiments with the dynamics of the Riemann zeta …
This theorem is based on holomorphy of studied functions and the fact that near a singularity point the real part of some rational function can take an arbitrary preassigned value.
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…
Continuing previous study of the Beurling zeta function, here we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the…
For an arbitrary complex number $a\neq 0$ we consider the distribution of values of the Riemann zeta-function $\zeta$ at the $a$-points of the function $\Delta$ which appears in the functional equation $\zeta(s)=\Delta(s)\zeta(1-s)$. These…
The evaluation of the Riemann zeta function at negative integers is a classical result typically obtained through analytic continuation or contour integration. In this paper, we present a novel and concise derivation of these special values…
The individual terms of the series representing the Riemann zeta function are examined geometrically from their accumulated plot in the complex plane. Symmetry is identified and determined mathematically for comparison with more traditional…
Assuming the Riemann hypothesis, we show that a certain vertical distribution of the nontrivial zeros of the Riemann zeta-function is equivalent to the generalized Riemann hypothesis for Dirichlet $L$-functions. Furthermore, under both the…
In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We first obtain two…
The Argand diagram is used to display some characteristics of the Riemann Zeta function. The zeros of the Zeta function on the complex plane give rise to an infinite sequence of closed loops, all passing through the origin of the diagram.…
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}…
We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary point process. We characterize those stationary processes with finite second moment for which, after…
The proof of the conjecture of the Birch and Swinnerton - Dyer is presented in the paper. The Riemann's hypothesis on the distribution of non-trivial zeroes of the zeta-function of Riemann, previously proven, is word to prove this…
The two-parameter series over the critical zeros of the Riemann Zeta function $Re\sum_{\rho}\frac{x^{(\rho-a)/4a}}{\sqrt{\rho-a}\sinh[\frac{\pi}{2}\sqrt{\frac{\rho-a}{a}}]\zeta'(\rho)}$ is evaluated in terms of $\zeta(s)$ on the real axis.
We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the…
We consider the PDE flow associated to Riemann zeta and general Dirichlet $L$-functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow…
We introduce new zeta functions related to an endomorphism $\phi$ of a discrete group $\Gamma$. They are of two types: counting numbers of fixed ($\rho\sim \rho\circ\phi^n$) irreducible representations for iterations of $\phi$ from an…
The critical line of the Riemann zeta function is studied from a new viewpoint. It is found that the ratio between the zeta function at any zero and the corresponding one at a conjugate point has a certain phase and its absolute value is…
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…
Results of a multipart work are outlined. Use is made therein of the conjunction of the Riemann hypothesis, RH, and hypotheses advanced by the author. Let z(n) be the nth nonreal zero of the Riemann zeta-function with positive imaginary…
We have done a statistical analysis of some properties of the contour lines Im$(\zeta (s))$ = 0 of the Riemann zeta function. We find that this function is broken up into strips whose average width on the critical line does not appear to…