Related papers: Fourier Analysis and the closed form for the Zeta …

Let X be a regular scheme, projective and flat over the integers. Let A be the constant in the conjectured functional equation for the zeta-function of X. We give a conjecture computing A in terms of Euler characteristics of derived…

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the…

We have looked at the evaluation of the Riemann Zeta function at odd arguments and have provided a simple formula to approximate the value with exponential convergence. We have compared it with various other formulae present in literature.…

For a pair of positive integers $n,k$ with $n\geq 2$, in this paper we prove that $$ \sum_{r=1}^k\sum_{|\bf\alpha|=k}{k\choose\bf\alpha} \zeta(n\bf\alpha)=\zeta(n)^k =\sum^k_{r=1}\sum_{|\bf\alpha|=k}…

In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.

As one of the asymptotic formulas for the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In 2003, R. Garunk\v{s}tis, A. Laurin\v{c}ikas, and J. Steuding (in [1]) proved the…

The zeta functions for the Schr\"odinger equation with a triangular potential are investigated. Values of the zeta functions are computed using both the Weierstrass factorization theorem and analytic continuation via contour integration.…

The classical inverse problem of recovering a simply connected smooth planar domain from the Steklov spectrum \cite{E} is equivalent to the problem of recovering, up to a conformal equivalence, a positive function $a\in C^\infty({\mathbb…

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of…

We provide the formula of motivic zeta function for semi-quasihomogeneous singularities and in dimension two, we determine the poles of zeta functions. We also give another formula for stringy E-function using embedded…

In this shortnote, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values $\:\zeta{(2 k +1)}$, $\zeta{(s)}$ being the Riemann zeta function and $k$ a positive integer, is…

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 evaluate the multiple zeta values $\zeta(\{2\}^k)$ by proving a certain factorization property. The proof uses a combinatorial bijection and elementary telescoping series. We show how the infinite product for the sine function in fact…

We provide an introduction of some basic facts of uniformly almost periodic functions, such as Fourier series representations. A result is then proved about Fourier coefficients which is a generalization of the purely periodic case. We then…

We establish the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relations between power series and trigonometric series,…

In this paper we study the Fourier coefficients of theta functions attached to Dirichlet characters at cusps other than infinity. The method is based on expressing them in terms of explicit elements of the adelic Schwartz space and studying…

We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the…

Let $\mathfrak{o}$ be a compact discrete valuation ring and $n\geq 2$. We introduce a method to study the cotype zeta function of subalgebras of $\mathfrak{o}^n$. This multivariable series encodes the number of finite-index subalgebras…

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…

This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…