Related papers: The High Precision Numerical Calculation of Stielt…

A code for the numerical evaluation of hyperelliptic theta-functions is presented. Characteristic quantities of the underlying Riemann surface such as its periods are determined with the help of spectral methods. The code is optimized for…

Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those…

The Stieltjes constants $\gamma_k(a)$ appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about its only pole at $s=1$. We generalize a technique of Addison for the Euler…

The note is a continuation of the previous paper ``On q-analogues of Riemann's zeta'' (math.QA/980499). It contains an output of the computer program calculating the zeros of the ``sharp'' q-zeta function.

In one of his posthumous papers, conserved in G\"ottingen, Riemann considers the derivatives of $\log\zeta(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some…

A detailed study of a double integral representation of the Catalan's constant allows us to identify a duality identity for the Stieltjes transform on which it is based. This duality identity is then extended to an arbitrary dimensional…

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The…

We consider sums of the form $\sum \phi(\gamma)$, where $\phi$ is a given function, and $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such…

The paper surveys the basic properties of generalized Stieltjes functions including some new ones. We introduce the notion of the exact Stieltjes order and give a criterion of exactness, simple sufficient conditions and some prototypical…

We consider the real part $\Re(\zeta(s))$ of the Riemann zeta-function $\zeta(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately the constant $\sigma_0 = 1.19\ldots$ which is defined to be the supremum of $\sigma$ such…

The Laurent Stieltjes constants $\gamma_n(\chi)$ are, up to a trivial coefficient, the coefficients of the Laurent expansion of the usual Dirichlet $L$-series: when $\chi$ is non principal, $(-1)^n\gamma_n(\chi)$ is simply the value of the…

In this paper new series for the first and second Stieltjes constants (also known as generalized Euler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the…

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions $\frac{1}{2} + i(a n + b)$. It reveals noticeable relation between the discrete moments and the continuous moment of the…

An elementary approach for computing the values at negative integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that…

We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with help of the Hermite polynomials. From the explicit formula…

We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums…

We present a large number of analytic evaluations of Euler sums, namely sums such as \begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \nonumber \end{align}…

This article gives a direct formula for the computation of B(n) using the asymptotic formula $$B (n) \approx 2 {\frac {n!}{{\pi}^{n}{2}^{n}}}$$ where n is even and $n >> 1$. This is simply based on the fact that $\zeta (n)$ is very near 1…

For $\chi_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,\chi_k)$, expressed as linear combinations of Hurwitz $\zeta$ functions, are found for $s=2,3$ and some…

We provide rapidly converging formulae for the Riemann zeta function at odd integers using the Lambert series $\mathscr{L}_q(s) = \sum_{n=1}^\infty n^{s} q^{n}/(1-q^n)$, $s=-(4k\pm 1)$. Our main formula for $\zeta(4k-1)$ converges at rate…