Related papers: A Non-Sieving Application of the Euler Zeta Functi…

The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line ${\rm Re}\,s=…

This article proves the Riemann hypothesis, which states that all non-trivial zeros of the zeta function have a real part equal to 1/2. We inspect in detail the integral form of the (symmetrized) completed zeta function, which is a product…

We prove certain conjecture holds true for a finite category which has M\"obius inversion. The conjecture states a relationship between the zeta function of a finite category and the Euler characteristic of a finite category.

We investigate the internal space of Bessel functions which is associated to the group Z of positive and negative integers defining their orders. As a result we propose and prove a new unifying formula (to be added to the huge literature on…

We introduce a new notion of $\boxast$-product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability and commuting with the limit of rational series. In the same…

The purpose of this article is to introduce the concept of invariance and its properties. These properties can be used to check the primality of a number. Combining these properties with the Euler theorem, it is possible to generalize this…

The absolute zeta function for a scheme $X$ of finite type over $\mathbb{Z}$ satisfying a certain condition is defined as the limit as $p\to 1$ of the congruent zeta function for $X\otimes\mathbb{F}_p$. In 2016, after calculating absolute…

We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the…

We discuss some formulae which express the Alexander polynomial (and thus the zeta-function of the classical monodromy transformation) of a plane curve singularity in terms of the ring of functions on the curve. One of them describes the…

To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special…

We will describe an algorithm to arrange all the positive and negative integer numbers. This array of numbers permits grouping them in six different Classes, $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, and $\zeta$. Particularly,…

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…

We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz…

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the…

A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla-Selberg series formula, for which an alternative proof is thereby…

We conjecture the full asymptotic expansion of a product of Riemann zeta functions, evaluated at the non-trivial zeros of the zeta function, with shifts added in each argument. By taking derivatives with respect to these shifts, we form a…

We show that for any fixed base $a$, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base $a$ expansion; the case $a=2$ was already established by Cohen-Selfridge…

The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain a local index formula for "abstract elliptic pseudodifferential operators"…

Various new identities, recurrence relations, integral representations, connection and explicit formulas are established for the Bernoulli, Euler numbers and the values of Riemann's zeta function. To do this, we explore properties of some…

By employing contour integration the derivation of a generalized double finite series involving the Hurwitz-Lerch zeta function is used to derive closed form formulae in terms of special functions. We use this procedure to find special…