Related papers: Experiments with zeta zeros and Perron's formula

We generalize our recent construction of the zeros of the Riemann $\zeta$-function to two infinite classes of $L$-functions, Dirichlet $L$-functions and those based on level one modular forms. More specifically, we show that there are an…

Ferrers diagrams are used to visually represent integer partitions. We describe a way to use Ferrers diagrams to uniquely represent integers in terms of their prime factors. This leads to a lower bound on the number of primes less than a…

We provide a lower bound for the dimension of the vector space spanned by 1 and by the values of the Riemann Zeta function at the first odd integers. As a consequence, the Zeta function takes infinitely many irrational values at odd…

A discussion involving the evaluation of the sum $\sum_{0<\gamma\le T} |\zeta(1/2+i\gamma)|^2$ is presented, where $\gamma$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Three theorems involving certain…

In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis.

In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta…

In this paper, we give a short elementary proof of the well known Euler's recurrence formula for the Riemann zeta function at positive even integers and integral representations of the Riemann zeta function at positive integers and at…

This paper considers some infinite series involving the Riemann zeta function.

In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative…

Number theory is an abstract mathematical field that has found a fertile environment for development in theoretical physics. In particular, several physical systems were related to the zeros of the Riemann-zeta function. In this work we…

In this paper, we show that Riemann hypothesis (concerning zeros of the zeta function in the critical strip) is equivalent to the analytic continuation of Euler products obtained by restricting the Euler zeta product to suitable subsets…

We rewrite Riemann Zeta function as a sum over the primes. Each term of the sum is a product that depends only on the summation index (a prime) and the primes following it.

Numerical investigations around a transformation of Landau's formula suggest certain statistical regularities in the distribution of zeros of the Riemann zeta function.

In this paper we study the higher-order Euler numbers and polynomials and we introduce the mutiple zeta functions which interpolate higher-order Euler polynomials and numbers at negative integers

Some statements concerning the distribution of imaginary parts of zeros of the Riemann zeta\,-function are established. These assertions are connected with so\,-called `Gram's law' or `Gram's rule'. In particular, we give a proof of several…

After a brief introduction to Ramanujan's method of summation, we give an expansion of the Riemann Zeta function in the critical strip as a convergent series $\sum_{m\geq 0}x_m P_m(s) $ where the functions $P_m$ are polynomials with their…

In this article, I derive a new approach to estimate the number of non-trivial zeros of a given Dedekind zeta function with absolute height at most $T\geq1$ counted with multiplicity. The error term in corresponding asymptotic formula…

The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann xi-function, $\xi(s)$. Thus, if the Riemann Hypothesis is true for the zeta-function, it is true for $\xi(s)$. Since $\xi(s)$ is entire, the zeros of…

Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent,…

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…