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 capabilities. 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, even though, lies within the incontrovertible fact that unique values of the Riemann zeta functionality may be written by utilizing Bernoulli numbers. This ends up in extra complex issues, a couple of that are handled during this booklet: old comments on Bernoulli numbers and the formulation for the sum of powers of consecutive integers; a formulation for Bernoulli numbers by means 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 features, and their detailed values at appropriate 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; category quantity formulation for confident certain binary quadratic kinds; congruences among a few type numbers and Bernoulli numbers; uncomplicated zeta capabilities of prehomogeneous vector areas; Hurwitz numbers; Barnes a number of zeta capabilities and their exact values; the practical equation of the doub
le zeta features; and poly-Bernoulli numbers. An appendix via Don Zagier on curious and unique identities for Bernoulli numbers can also be provided. This publication should be stress-free either for amateurs and for pro researchers. as the logical family members among the chapters are loosely hooked up, readers can commence with any bankruptcy counting on their pursuits. The expositions of the subjects aren't continuously commonplace, and a few elements are thoroughly new.
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