By Tsuneo Arakawa,Tomoyoshi Ibukiyama,Masanobu Kaneko,Don B. Zagier
Two significant matters are taken care of during this publication. 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 provide formulation for the sums of powers of consecutive integers. the true cause that they're crucial for quantity idea, in spite of the fact that, lies within the indisputable fact that exact values of the Riemann zeta functionality may be written by utilizing Bernoulli numbers. This ends up in extra complex themes, a few that are taken care of during this ebook: ancient 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 features, and their unique values at compatible integers; a number of formulation of exponential sums expressed via generalized Bernoulli numbers; the relation among excellent sessions of orders of quadratic fields and equivalence periods of binary quadratic varieties; category quantity formulation for optimistic yes binary quadratic types; congruences among a few type numbers and Bernoulli numbers; basic zeta services of prehomogeneous vector areas; Hurwitz numbers; Barnes a number of zeta services and their precise values; the practical 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 can be provided. This booklet could be stress-free either for amateurs and for pro researchers. as the logical family among the chapters are loosely hooked up, readers can begin with any bankruptcy looking on their pursuits. The expositions of the themes will not be continually normal, and a few elements are thoroughly new.
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Bernoulli Numbers and Zeta Functions (Springer Monographs in Mathematics) by Tsuneo Arakawa,Tomoyoshi Ibukiyama,Masanobu Kaneko,Don B. Zagier