Mathematisch-naturwissenschaftlichen Fakultät Doctor of Philosophy Twisted Second Moments and Explicit Formulae of the Riemann Zeta-Function by Nicolas Martinez Robles Verschiedene Aspekte, die analytische Zahlentheorie und die Riemann zeta-Funktion verbinden, werden erweitert. Dies beinhaltet: 1. explizite Formeln, die eine Verbindung zwischen der Möbiusfunktion und den nichttrivialen Nullstel...

LITP 4 place Jussieu 75005 Paris Abs t rac t Motivated by symbolic dynamics and algebraic geometry over finite fields, we define cyclic languages and the zeta function of a language. The main result is that the zeta function of a cyclic language which is recognizable by a finite automaton is rational. I. In t roduct ion Motivated by algebraic geometry over finite fields and symbolic dynamics, w...

In this chapter, we will see a proof of the analytic continuation of the Riemann zeta function ζ(s) and the Dirichlet L function L(s, χ) via the Hurwitz zeta function. This then gives rise to a functional equation for ζ(s) and a direct computation for the value of this function at negative integer points. 1. The Hurwitz zeta function We have already seen the definition of the Riemann zeta funct...

A weak version of the Ihara formula is proved for zeta functions attached to quotients of the Bruhat-Tits building of PGL3. This formula expresses the zeta function in terms of Hecke-Operators. It is the first step towards an arithmetical interpretation of the combinatorially defined zeta function.

The zeta family includes zeta, eta, and FcepsilonRIgamma (Fcgamma). Dimers of the zeta family proteins function as signal transducing subunits of the T cell antigen receptor (TCR), the pre-TCR, and a subset of Fc receptors. In mice lacking zeta/eta chains, T cell development is impaired, yet low numbers of CD4+ and CD8+ T cells develop. This finding suggests either that pre-TCR and TCR complexe...

We introduce and study elliptic zeta values, a two-parameter deformation of the values of Riemann’s zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share the SL(3,Z) modular properties of this function. Elliptic zeta values at even integers are related to Eisenstein series and thus to sums of odd powers of divisor...

This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators {Tm : m ≥ 1} given by Tm(f )(a, c) = 1 m ∑m−1 k=0 f ( a+k m ,mc) acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. The actions of various related operators on these function spaces are de...

Introduced by Weil, the p-adic zeta function associated to a polynomial f over Zp was systematically studied by Igusa in the non-archimedean wing of his theory of local zeta functions, which also includes archimedean (real and complex) zeta functions [18][19]. The p-adic zeta function is a meromorphic function on the complex plane, and contains information about the number of solutions of the c...

This article improves the estimate of the size of the definite integral of S(t), the argument of the Riemann zeta-function. The primary application of this improvement is Turing’s Method for the Riemann zeta-function. Analogous improvements are given for the arguments of Dirichlet L-functions and of Dedekind zeta-functions.

It is well-known that the Euler product for the Riemann zeta function ζ(s) is still valid for !(s) = 1 and s "= 1. In this paper, we extend this result to zeta functions of number fields. In particular, the Dedekind zeta function ζk(s) for any algebraic number field k and the Hecke zeta function ζ(s,χ) for the rational number field are shown to have the Euler product on the line !(s) = 1 except...