Suppose Y is a regular covering of a graph X with covering transformation group π = Z. This paper gives an explicit formula for the L zeta function of Y and computes examples. When π = Z, the L zeta function is an algebraic function. As a consequence it extends to a meromorphic function on a Riemann surface. The meromorphic extension provides a setting to generalize known properties of zeta fun...

Suppose Y is a regular covering of a graph X with covering transformation group π = Z. This paper gives an explicit formula for the L2 zeta function of Y and computes examples. When π = Z, the L2 zeta function is an algebraic function. As a consequence it extends to a meromorphic function on a Riemann surface. The meromorphic extension provides a setting to generalize known properties of zeta f...

[This document is http://www.math.umn.edu/ ̃garrett/m/complex/notes 2014-15/09c Riemann and zeta.pdf] 1. Riemann’s explicit formula 2. Analytic continuation and functional equation of ζ(s) 3. Appendix: Perron identity [Riemann 1859] exhibited a precise relationship between primes and zeros of ζ(s). A similar idea applies to any zeta or L-function with analytic continuation, functional equation, ...

0 |ζ( 1 2 + it)| dt. For positive real numbers k, it is believed that Mk(T ) ∼ CkT (logT ) 2 for a positive constant Ck. A precise value for Ck was conjectured by Keating and Snaith [9] based on considerations from random matrix theory. Subsequently, an alternative approach, based on multiple Dirichlet series and producing the same conjecture, was given by Diaconu, Goldfeld and Hoffstein [4]. R...

c © W W L Chen, 1990, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage an...

This paper provides a proof of Dirichlet’s theorem, which states that when (m,a) = 1, there are infinitely many primes p such that p ≡ a (mod m). The proof explained here mirrors Serre’s proof in [1]. However, one distinction is that the Riemann zeta function is used as motivation to obtain results for the Dirichlet L-functions, which yields the above result. In fact, this paper can be read fro...

On the Riemann hypothesis we establish a uniform upper estimate for zeta(s)/zeta (s + A), 0 < or = A, on the critical line. We use this to give a purely complex-analytic variant of Báez-Duarte's proof of a strengthened Nyman-Beurling criterion for the validity of the Riemann Hypothesis. We investigate function-theoretically some of the functions defined by Báez-Duarte in his study and we show t...

We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants γk(a). These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials Ln to investigate a certain sum Sγ(n) involv...

“Fuzzy” models and 802.11 mesh networks have garnered improbable interest from both computational biologists and experts in the last several years. Given the current status of real-time epistemologies, futurists daringly desire the construction of DHCP, which embodies the practical principles of networking. In order to accomplish this ambition, we introduce new concurrent methodologies (Krang),...