Ramanujan’s notebooks, by Bruce C. Berndt. Part 1. Pp 357. DM188. 1985. ISBN 3-540-96110-0 (Springer)

Renormalization of multiple zeta values

One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Wojciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean

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Artin's primitive root conjecture -a survey -

Values of Euler-Zagier's multiple zeta function at non-positive integers are studied, especially at (0,0,...,−n) and (−n,0,...,0). Further we prove a symmetric formula among values at non-positive integers.

/pdf/multiple-zeta-values-at-non-positive-integers-232l6nj12b.pdf

Multiple Zeta Values at Non-Positive Integers

Chapter 1 Flat Space Fourier Analysis on R^m- 11 Distributions or Generalized Functions- 12 Fourier Integrals- 13 Fourier Series and the Poisson Summation Formula- 14 Mellin Transforms, Epstein and Dedekind Zeta Functions- 15 Finite Symmetric Spaces, Wavelets, Quasicrystals, Weyl's Criterion for Uniform Distribution- Chapter 2 A Compact Symmetric Space--The Sphere- 21 Fourier Analysis on the Sphere- 22 O(3) and R^3 The Radon Transform- Chapter 3 The Poincare Upper Half-Plane- 31 Hyperbolic Geometry- 32 Harmonic Analysis on H- 33 Fundamental Domains for Discrete Subgroups ? of G = SL(2, R)- 34 Modular of Automorphic Forms--Classical- 35 Automorphic Forms--Not So Classical--Maass Waveforms- 36 Modular Forms and Dirichlet Series Hecke Theory and Generalizations- References- Index

Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Analytic continuation of the multiple zeta-function is established by a simple application of the Euler-Maclaurin summation formula. Multiple zeta values at non-positive integers are defined and their properties are investigated.

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Analytic continuation of multiple zeta-functions and their values at non-positive integers

Convolutions of the von mangoldt function and related dirichlet series

In this paper, we first give a brief survey on the theory of meromorphic continuation and natural boundaries of multiple Dirichlet series. Then we consider the double Dirichlet series Φ2(s) defined by the convolution of logarithmic derivatives of the Riemann zeta-function. Especially we propose the conjecture that Φ2(s) would have the natural boundary on <s = 1, and give a supportive evidence. We further present an application of Φ2(s) to the Riesz mean, and discuss its multiple analogues. 1. The analytic continuation of multiple Dirichlet series Let s = σ+it be a complex variable, and P (X1, . . . , Xr) a polynomial of complex coefficients. The multiple zeta-function ζr(s; P ) = ∞

Conrelutions of the von Mangoldt function and related Dirichlet series (英文)

© Foundation Compositio Mathematica, 1995, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Shigeki Egami

Papers

Analytic continuation of multiple zeta-functions and their values at non-positive integers

Convolutions of the von mangoldt function and related dirichlet series

Conrelutions of the von Mangoldt function and related Dirichlet series (英文)

An elliptic analogue of the multiple Dedekind sums

An elliptic analogue of the multiple dedekind sums (英文)