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

Two significant matters are taken care of during this booklet. the most one is the speculation of Bernoulli numbers and the opposite is the speculation of zeta services. traditionally, Bernoulli numbers have been brought to offer formulation for the sums of powers of consecutive integers. the genuine cause that they're quintessential for quantity thought, in spite of the fact that, lies within the indisputable fact that unique values of the Riemann zeta functionality may be written through the use of Bernoulli numbers. This ends up in extra complex subject matters, a few that are taken care of during this e-book: ancient comments 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 similar idea of p-adic measures; the Euler–Maclaurin summation formulation; the practical equation of the Riemann zeta functionality and the Dirichlet L services, and their specified values at compatible integers; a number of formulation of exponential sums expressed by means of generalized Bernoulli numbers; the relation among excellent periods of orders of quadratic fields and equivalence periods of binary quadratic varieties; classification quantity formulation for optimistic sure binary quadratic kinds; congruences among a few classification numbers and Bernoulli numbers; basic zeta features of prehomogeneous vector areas; Hurwitz numbers; Barnes a number of zeta services and their exact values; the useful equation of the doub

le zeta features; and poly-Bernoulli numbers. An appendix by means of Don Zagier on curious and unique identities for Bernoulli numbers is additionally provided. This e-book might be relaxing either for amateurs and for pro researchers. as the logical relatives among the chapters are loosely attached, readers can commence with any bankruptcy looking on their pursuits. The expositions of the subjects are usually not continually normal, and a few components are thoroughly new.

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