Masanobu Kaneko

Tomoyoshi Ibukiyama

Tsuneo Arakawa

Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen–von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler–Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; class number formula for positive definite binary quadratic forms; congruences between some class numbers and Bernoulli numbers; simple zeta functions of prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta functions and their special values; the functional equation of the doub le zeta functions; and poly-Bernoulli numbers. An appendix by Don Zagier on curious and exotic identities for Bernoulli numbers is also supplied. This book will be enjoyable both for amateurs and for professional researchers. Because the logical relations between the chapters are loosely connected, readers can start with any chapter depending on their interests. The expositions of the topics are not always typical, and some parts are completely new.

Bernoulli Numbers and Zeta Functions

Lin Weng

This invaluable volume collects papers written by many of the world''s top experts on L -functions. It not only covers a wide range of topics from algebraic and analytic number theories, automorphic forms, to geometry and mathematical physics, but also treats the theory as a whole. The contributions reflect the latest, most advanced and most important aspects of L- functions. In particular, it contains Hida''s lecture notes at the conference and at the Eigenvariety semester in Harvard University and Weng''s detailed account of his works on high rank zeta functions and non-abelian L -functions. Sample Chapter(s). Chapter 1: Quantum Maass Forms (435 KB). Contents: Quantum Maass Forms (R Bruggeman); o-invariant of p -Adic L -Functions (H Hida); Siegel Modular Forms of Weight Three and Conjectural Correspondence of Shimura Type and Langlands Type (T Ibukiyama); Convolutions of Fourier Coefficients of Cusp Forms and the Circle Method (M Jutila); On an Extension of the Derivation Relation for Multiple Zeta Values (M Kaneko); On Symmetric Powers of Cusp Forms on GL 2 (H H Kim); Zeta Functions of Root Systems (Y Komori et al.); Sums of Kloosterman Sums Revisted (Y Motohashi); The LindelAf Class of L -Functions (K Murty); A Proof of the Riemann Hypothesis for the Weng Zeta Function of Rank 3 for the Rationals (M Suzuki); Elliptic Curves Arising from the Spectral Zeta Function for Non-Commutative Harmonic Oscillators and o 0 (4)-Modular Forms (K Kimoto & M Wakayama); A Geometric Approach to L -Functions (L Weng). Readership: Graduate students, lecturers, and active researchers in various branches of mathematics, such as algebra, analysis, geometry and mathematical physics."

Masanobu Kaneko

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