# Bernoulli Numbers and Zeta Functions (Springer Monographs in by Tsuneo Arakawa,Tomoyoshi Ibukiyama,Masanobu Kaneko,Don B. PDF

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

Two significant topics are handled during this publication. the most one is the idea of Bernoulli numbers and the opposite is the speculation of zeta features. traditionally, Bernoulli numbers have been brought to provide formulation for the sums of powers of consecutive integers. the genuine cause that they're critical for quantity conception, notwithstanding, lies within the indisputable fact that specific values of the Riemann zeta functionality might be written by utilizing Bernoulli numbers. This results in extra complicated subject matters, a couple of that are handled during this ebook: 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 similar conception of *p*-adic measures; the Euler–Maclaurin summation formulation; the sensible equation of the Riemann zeta functionality and the Dirichlet L capabilities, and their exact values at compatible integers; quite a few 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 varieties; classification quantity formulation for optimistic yes binary quadratic varieties; congruences among a few category numbers and Bernoulli numbers; easy zeta services of prehomogeneous vector areas; Hurwitz numbers; Barnes a number of zeta services and their particular values; the useful equation of the doub

le zeta services; and poly-Bernoulli numbers. An appendix via Don Zagier on curious and unique identities for Bernoulli numbers is usually provided. This e-book could be relaxing either for amateurs and for pro researchers. as the logical family among the chapters are loosely hooked up, readers can commence with any bankruptcy reckoning on their pursuits. The expositions of the subjects aren't continuously common, 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

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