Related papers: Zeros of $\mathop{\mathcal R}(s)$ on the fourth qu…

We prove that the number of zeros $\varrho=\beta+i\gamma$ of $\mathop{\mathcal R}(s)$ with $0<\gamma\le T$ is given by \[N(T)=\frac{T}{4\pi}\log\frac{T}{2\pi}-\frac{T}{4\pi}-\frac12\sqrt{\frac{T}{2\pi}}+O(T^{2/5}\log^2 T).\] Here…

We have computed all zeros $\beta+i\gamma$ of $\mathop{\mathcal R }(s)$ with $0<\gamma<215946.3$. A total of 162215 zeros with 25 correct decimal digits. In this paper we offer some statistic based on this set of zeros. Perhaps the main…

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…

Let $F$ be a non-degenerate quadratic form on an $n$-dimensional vector space $V$ over the rational numbers. One is interested in counting the number of zeros of the quadratic form whose coordinates are restricted in a smoothed box of size…

In this paper we study the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. The zeros of these polynomials are the nodes for complex Gaussian…

It is proved that $s=-2n$ is a simple zero of $\mathop{\mathcal R}(s)$ for each integer $n\ge1$. Here $\mathop{\mathcal R}(s)$ is the function found by Siegel in Riemann's posthumous papers.

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree $n$, with not necessarily independent coefficients, decays like $\sqrt{\log n/n}$. Our proofs rely on…

A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective…

To define an explicit regions without zeros of $\mathop{\mathcal R}(s)$, in a previous paper we obtained an approximation to $\mathop{\mathcal R}(s)$ of type $f(s)(1+U)$ with $|U|< 1$. But this $U$ do not tend to zero when $t\to+\infty$. In…

For the generalized Jacobi, Laguerre and Hermite polynomials $P_n^{(\alpha_n, \beta_n)} (x), L_n^{(\alpha_n)} (x),$\break $H_n^{(\gamma_n)} (x)$ the limit distributions of the zeros are found, when the sequences $\alpha_n$ or $\beta_n$ tend…

We study the zero distribution of non-orthogonal polynomials attached to $g(n)=s(n)=n^2$: \begin{equation*} Q_n^g(x)= x \sum_{k=1}^n g(k) \, Q_{n-k}^g(x), \quad Q_0^g(x):=1. \end{equation*} It is known that the case $g=id$ involves…

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{(\alpha_n, \beta_n)}$ is studied, assuming that $$ \lim_{n\to\infty} \frac{\alpha_n}{n}=A, \qquad \lim_{n\to\infty} \frac{\beta _n}{n}=B, $$ with…

Numerical data suggest that the zeros $\rho$ of the auxiliary Riemann function in the upper half-plane satisfy $\mathop{\mathrm{Re}}(\rho)<1$. We show that this is true for those zeros with $\mathop{\mathrm{Im}}(\rho)> 3.9211\dots10^{65}$.…

We numerically investigate, for zeros $\rho=1/2+i\gamma$, the statistics of the imaginary part of $\log(\zeta^\prime(1/2+i\gamma))$, computed by continuous variation along a vertical line from $\sigma=4$ to $4+i\gamma$ and then along a…

A fresh approach to the long debated question is proposed, starting from the GRAM-BACKLUND analytical continuation of the Zeta function (G-B Zeta expression). Consideration is given to the symmetric (even-exponent) and anti-symmetric (odd…

We study the horizontal distribution of zeros of $\zeta'(s)$ which are denoted as $\rho'=\beta'+i\gamma'$. We assume the Riemann hypothesis which implies $\beta'\geqslant1/2$ for any non-real zero $\rho'$, equality being possible only at a…

We consider Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ with varying negative parameters $\alpha_n$, such that the limit $A = -\lim_n \alpha_n/n$ exists and belongs to $(0,1)$. For $A > 1$, it is known that the zeros accumulate along an…

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.

For any $\varepsilon >0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert \alpha n + \beta \rVert < x^{-\frac{1}{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real number $x$. In…

We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these…