By Tsuneo Arakawa,Tomoyoshi Ibukiyama,Masanobu Kaneko,Don B. Zagier

Two significant topics are handled during this booklet. the most one is the idea of Bernoulli numbers and the opposite is the speculation of zeta services. traditionally, Bernoulli numbers have been brought to provide formulation for the sums of powers of consecutive integers. the true cause that they're crucial for quantity idea, although, lies within the indisputable fact that specified values of the Riemann zeta functionality could be written through the use of Bernoulli numbers. This results in extra complicated themes, a few that are taken care of during this publication: old feedback on Bernoulli numbers and the formulation for the sum of powers of consecutive integers; a formulation for Bernoulli numbers by way of Stirling numbers; the Clausen–von Staudt theorem at the denominators of Bernoulli numbers; Kummer's congruence among Bernoulli numbers and a comparable idea of *p*-adic measures; the Euler–Maclaurin summation formulation; the sensible equation of the Riemann zeta functionality and the Dirichlet L services, and their exact values at compatible integers; a number of formulation of exponential sums expressed by way of generalized Bernoulli numbers; the relation among excellent periods of orders of quadratic fields and equivalence sessions of binary quadratic types; classification quantity formulation for optimistic certain binary quadratic types; congruences among a few type numbers and Bernoulli numbers; easy zeta features of prehomogeneous vector areas; Hurwitz numbers; Barnes a number of zeta features and their certain values; the sensible equation of the doub

le zeta services; and poly-Bernoulli numbers. An appendix through Don Zagier on curious and unique identities for Bernoulli numbers can be provided. This ebook might be relaxing either for amateurs and for pro researchers. as the logical kinfolk among the chapters are loosely hooked up, readers can commence with any bankruptcy counting on their pursuits. The expositions of the subjects are usually not regularly commonplace, and a few components are thoroughly new.

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