The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

An Analytical and Numerical Detour for the Riemann Hypothesis

From the functional equation of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function and its family of associated functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the function. This family is a mathematical and numerical tool which makes it possible to estimate the value of the function at a point in the critical strip from a point on the critical line .Generating estimates of at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the and functions over finite fields.

Bounding the log-derivative of the zeta-function

Assuming the Riemann hypothesis we establish explicit bounds for the modulus of the log-derivative of Riemann’s zeta-function in the critical strip.

From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results

A century ago Srinivasa Ramanujan --- the great self-taught Indian genius of mathematics --- died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, $\sum_{n \geq 1} n$ and $\sum_{n \geq 1} n^{3}$. These values are sensible, however, as analytic continuations, which correspond to Riemann's $\zeta$-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory.

The Two-Layer Hierarchical Distribution Model of Zeros of Riemann’s Zeta Function along the Critical Line

This article numerically analyzes the distribution of the zeros of Riemann’s zeta function along the critical line (CL). The zeros are distributed according to a hierarchical two-layered model, one deterministic, the other stochastic. Following a complex plane anamorphosis involving the Lambert function, the distribution of zeros along the transformed CL follows the realization of a stochastic process of regularly spaced independent Gaussian random variables, each linked to a zero. The value of the standard deviation allows the possible overlapping of adjacent realizations of the random variables, over a narrow confidence interval. The hierarchical model splits the ζ function into sequential equivalence classes, with the range of probability densities of realizations coinciding with the spectrum of behavioral styles of the classes. The model aims to express, on the CL, the coordinates of the alternating cancellations of the real and imaginary parts of the ζ function, to dissect the formula for the number of zeros below a threshold, to estimate the statistical laws of two consecutive zeros, of function maxima and moments. This also helps explain the absence of multiple roots.

An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann’s ξs function and thus ζs indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates ζs anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, both an analytic expression for ζσ+it, everywhere inside the asymptotic t⟶∞ critical strip, as well as an approximate solution can be obtained, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of ζσ+it for different values of σ and equal values of t; this is illustrated in a number of figures.