Related papers: Riemann's Zeta Function: The alternating Xi-Functi…

It is known by a formula of Hasse-Sondow that the Riemann zeta function is given, for any $ s=\sigma+it \in \mathbb{C}$, by $ \sum_{n=0}^{\infty} \widetilde{A}(n,s)$ where $$ \widetilde{A}(n,s):=\frac{1}{2^{n+1}(1-2^{1-s})} \sum_{k=0}^n…

The usual Riemann-Siegel Z(t) is a real-valued function. We construct a complex function depending from t and from distance from critical line. It is linked to Riemann Xi(s) function by the same real scaling factor of the usual…

Using elementary methods we find surprising connections between the values of the Riemann Zeta Function over integers and the fractional parts of rational powers, and a connection between the Riemann Zeta Function and the Prime Zeta…

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.

Contour integral representations for Riemann's Zeta function and Dirichelet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800's, but somehow they do not…

For any $a\in\mathbb{C}$, the zeros of $\zeta(s)-a$, denoted by $\rho_a=\beta_a+i\gamma_a$, are called $a$-points of the Riemann zeta function $\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\zeta(s)$…

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

Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We…

The secondary zeta function $Z(s)=\sum_{n=1}^\infty\alpha_n^{-s}$, where $\rho_n=\frac12+i\alpha_n$ are the zeros of zeta with $\Im(\rho)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis…

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the…

Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin…

The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented.

The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of…

The individual terms of the series representing the Riemann zeta function are examined geometrically from their accumulated plot in the complex plane. Symmetry is identified and determined mathematically for comparison with more traditional…

In article, we explore the secondary zeta function $Z(s)$, which is defined as a generalized zeta type of series over imaginary parts of non-trivial zeros of the Riemann zeta function $\zeta(s)$. This function has been analytically…

Suppose $Y$ is a regular covering of a graph $X$ with covering transformation group $\pi = \mathbb{Z}$. This paper gives an explicit formula for the $L^2$ zeta function of $Y$ and computes examples. When $\pi = \mathbb{Z}$, the $L^2$ zeta…

This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…

The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions and more generally the same can be done with polylogarithms, for which several zeta functions are a special case. An…

We show that there is a contradiction between the Riemann's Hypothesis and some form of the theorem on the universality of the zeta function.