J. A. de Reyna · wovepaper

activity

20032026

most citedDynamical zeta functions and Kummer congruences

13 citations · 40 across the 43 of their papers we have counts for

collaborators

45 papers

Human Exposition of Claude's 2/3 Theorem on the zeros of zeta

J. Arias de Reyna

We expose Claude's proof that $N_{0} (T) \geq 0.6725 N (T)$ where $N (T)$ is the number of zeros of zeta satisfying $0 < Re ρ \leq T$ , and $N_{0} (T)$ the number of those zeros…

Riemann and the logarithmic derivatives of zeta

J. Arias de Reyna

In one of his posthumous papers, conserved in Göttingen, Riemann considers the derivatives of $lo g ζ (s)$ at the point $1/2$ , giving explicit values for them. Around 2010 we shared…

Statistic of zeros of Riemann auxiliary function

J. Arias de Reyna

We have computed all zeros $β + iγ$ of $R (s)$ with $0 < γ < 215946.3$ . A total of 162215 zeros with 25 correct decimal digits. In this paper we offer some statistic ba…

Report on some papers related to the function $R (s)$ found by Siegel in Riemann's posthumous papers

J. Arias de Reyna

In a letter to Weierstrass Riemann asserted that the number $N_{0} (T)$ of zeros of $ζ (s)$ on the critical line to height $T$ is approximately equal to the total number of zeros to th…

Simple bounds for the auxiliary function of Riemann

Juan Arias de Reyna

We give simple numerical bounds for $ζ (s)$ , $ϑ (s)$ , $R (s)$ , $Z (t)$ , for use in the numerical computation of these functions. The purpose of the paper is t…

A naive integral

Juan Arias de Reyna

In arXiv:2406.0243 two real functions $g (x, t)$ and $f (x, t)$ are defined, so that the Riemann-Siegel $Z$ function is given as \[Z(t)=\mathop{\mathrm{Re}}\Bigl\{\frac{u(t)e^{\frac{πi…

Levinson Functions

Juan Arias de Reyna

Starting from some of Norman Levinson's results, we construct interesting examples of functions $f (s)$ such that for $s = \frac{1}{2} + i t$ , we have $Z(t)=2\Re\{π^{-\frac{s}{2}}Γ(s/2)f(s)\…

Explicit van der Corput's $d$ -th derivative estimate

Juan Arias de Reyna

We give an explicit version for van der Corput's $d$ -th derivative estimate of exponential sums. $Theorem.$ Let $X$ , and $Y \in R$ be such that $\lfloor Y\rfloor>d…

Integral Representations of Riemann auxiliary function

Juan Arias de Reyna

We prove that the auxiliary function $R (s)$ has the integral representation \[\mathop{\mathcal R}(s)=-\frac{2^s π^{s}e^{πi s/4}}{Γ(s)}\int_0^\infty y^{s}\frac{1-e…

An Integral representation of $R (s)$ due to Gabcke

Juan Arias de Reyna

Gabcke proved a new integral expression for the auxiliary Riemann function \[\mathop{\mathcal R}(s)=2^{s/2}π^{s/2}e^{πi(s-1)/4}\int_{-\frac12\searrow\frac12} \frac{e^{-πi u^2/2+πi…

An entire function defined by Riemann

Juan Arias de Reyna

In one of the sheets in Riemann's Nachlass he defines an entire function and connect it with his zeta function. As in many pages in his Nachlass, Riemann is not giving complete pro…

Integral Representation for Riemann-Siegel $Z (t)$ function

Juan Arias de Reyna

We apply Poisson formula for a strip to give a representation of $Z (t)$ by means of an integral. \[F(t)=\int_{-\infty}^\infty \frac{h(x)ζ(4+ix)}{7\coshπ\frac{x-t}{7}}\,dx, \qquad Z…

Approximate formula for $Z (t)$

Juan Arias de Reyna

The series for the zeta function does not converge on the critical line but the function \[G(t)=\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{t}{2πn^2+t}\] satisfies $Z(t)=2\Re\{…

An expression for Riemann Siegel function

Juan Arias de Reyna

There are many analytic functions $U (t)$ satisfying $Z (t) = 2ℜ {e^{i ϑ (t)} U (t)}$ . Here, we consider an entire function $L (s)$ such that $U(t)=\…

On the approximation of the zeta function by Dirichlet polynomials

Juan Arias de Reyna

We prove that for $s = σ + i t$ with $σ \geq 0$ and $0 < t \leq x$ , we have \[ζ(s)=\sum_{n\le x}n^{-s}+\frac{x^{1-s}}{(s-1)}+Θ\frac{29}{14} x^{-σ},\qquad \frac{29}{14}=2.07142\dots\] where $Θ$ …

Density theorems for Riemann's auxiliary function

Juan Arias de Reyna

We prove a density theorem for the auxiliar function $R (s)$ found by Siegel in Riemann papers. Let $α$ be a real number with $\frac{1}{2} < α \leq 1$ , and let $N (α, T)$ b…

Mean Values of the auxiliary function

Juan Arias de Reyna

Let $R (s)$ be the function related to $ζ (s)$ found by Siegel in the papers of Riemann. In this paper we obtain the main terms of the mean values \[\frac{1}{T}\int…

Infinite Product of the Riemann auxiliary function

Juan Arias de Reyna

We obtain the product for the auxiliary function $R (s)$ and study some related functions as its phase $ω (t)$ at the critical line. The function $ω (t)$ determines…

Zeros of $R (s)$ on the fourth quadrant

Juan Arias de Reyna

We show that there is a sequence of zeros of $R (s)$ in the fourth quadrant. We show that the $n$ -th zero $ρ_{- n} = β_{- n} + i γ_{- n}$ , with $β_{- n} \sim 4 π^{2} n / lo g^{2} n$ …

Trivial zeros of Riemann auxiliary function

Juan Arias de Reyna

It is proved that $s = - 2 n$ is a simple zero of $R (s)$ for each integer $n \geq 1$ . Here $R (s)$ is the function found by Siegel in Riemann's posthumo…

On the number of zeros of $R (s)$

Juan Arias de Reyna

We prove that the number of zeros $ϱ = β + iγ$ of $R (s)$ with $0 < γ \leq T$ is given by \[N(T)=\frac{T}{4π}\log\frac{T}{2π}-\frac{T}{4π}-\frac12\sqrt{\frac{T}{2π}}…

On Siegel results about the zeros of the auxiliary function of Riemann

Juan Arias de Reyna

We state and give complete proof of the results of Siegel about the zeros of the auxiliary function of Riemann $R (s)$ . We point out the importance of the determin…

Riemann's Auxiliary Function. Right limit of zeros

Juan Arias de Reyna

Numerical data suggest that the zeros $ρ$ of the auxiliary Riemann function in the upper half-plane satisfy $Re (ρ) < 1$ . We show that this is true for those zeros w…

Note on the asymptotic of the auxiliary function

Juan Arias de Reyna

To define an explicit regions without zeros of $R (s)$ , in a previous paper we obtained an approximation to $R (s)$ of type $f (s) (1 + U)$ with $|U|<…