Related papers: The functional equation for $\zeta$

We give a functional equation for the refined Herglotz-Zagier function. It is analogous to a result in the theory of modular forms.

We prove a formula for the Mangoldt function which relates it to a sum over all the non-trivial zeros of the Riemann zeta function, in addition we analize a truncated version of it.

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

In this manuscript, we give effective methods for computing the zeta function of maximal orders on surfaces.

This paper contains a small improvement to the explicit bounds on the growth of the function $S(T)$. It is shown how more substantial improvements are possible if one has better explicit bounds on the growth of $|\zeta(\frac{1}{2}+it)|$.

The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation…

We give simple numerical bounds for $\zeta(s)$, $\vartheta(s)$, $\mathop{\mathcal R}(s)$, $Z(t)$, for use in the numerical computation of these functions. The purpose of the paper is to give bounds for several functions needed in the…

In this paper, we prove some inequalities for the differences and ratios of the beta function.

The paper reviews existing results about the statistical distribution of zeros for the three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of…

It is shown that Weng's zeta functions associated with arbitrary semisimple algebraic groups defined over the rational number field and their maximal parabolic subgroups satisfy the functional equations.

New recursion relations for the Riemann zeta function are introduced. Their derivation started from the standard functional equation. The new functional equations have both real and imaginary increment versions and can be applied over the…

We obtain a new proof of Hurwitz's formula for the Hurwitz zeta function $\zeta(s, a)$ beginning with Hermite's formula. The aim is to reveal a nice connection between $\zeta(s, a)$ and a special case of the Lommel function $S_{\mu,…

This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation…

We state and prove a function field analogue of Furusho for multiple zeta values.

We intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also…

This is primarily an overview article on some results and problems involving the classical Hardy function $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s). $$ In particular, we discuss the first…

In this article, we study the multiple zeta functions (MZF) and some of its variants at identical arguments. Using the harmonic product, these functions can be expressed as polynomials in the Riemann zeta function. Firstly, we note that an…

We define a divisorial motivic zeta function for stable curves with marked points which agrees with Kapranov's motivic zeta function when the curve is smooth and unmarked. We show that this zeta function is rational, and give a formula in…

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