By Tsuneo Arakawa, Visit Amazon's Tomoyoshi Ibukiyama Page, search results, Learn about Author Central, Tomoyoshi Ibukiyama, , Masanobu Kaneko, Don B. Zagier
Two significant topics are taken care of during this booklet. the most one is the idea of Bernoulli numbers and the opposite is the idea of zeta services. traditionally, Bernoulli numbers have been brought to provide formulation for the sums of powers of consecutive integers. the genuine cause that they're imperative for quantity conception, although, lies within the incontrovertible fact that particular values of the Riemann zeta functionality may be written through the use of Bernoulli numbers. This results in extra complicated issues, a couple of that are handled during this e-book: 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 exact values at compatible integers; numerous formulation of exponential sums expressed by means of generalized Bernoulli numbers; the relation among excellent sessions of orders of quadratic fields and equivalence periods of binary quadratic types; type quantity formulation for optimistic yes binary quadratic varieties; congruences among a few category numbers and Bernoulli numbers; basic zeta features of prehomogeneous vector areas; Hurwitz numbers; Barnes a number of zeta features and their detailed values; the sensible equation of the double zeta features; and poly-Bernoulli numbers. An appendix through Don Zagier on curious and unique identities for Bernoulli numbers can be provided. This ebook may be relaxing either for amateurs and for pro researchers. as the logical kin among the chapters are loosely hooked up, readers can commence with any bankruptcy looking on their pursuits. The expositions of the themes should not regularly average, and a few elements are thoroughly new.
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Extra info for Bernoulli Numbers and Zeta Functions
1 Theorem of Clausen and von Staudt The denominators of the Bernoulli numbers can be completely determined. This is due to Clausen1  and von Staudt2 . More precisely, the “fractional part” of Bn is given by the following theorem. This result gives a foundation for studying p-adic properties of the Bernoulli numbers. It also plays a fundamental role in the theory of p-adic modular forms through the Eisenstein3 series . 1. Cn is an integer/: p 1jn Here, the sum runs over all the prime numbers p such that p 1 divides n.
We adopt Knuth’s notation . 1 (Stirling numbers of the second kind (Stirling’s subset numbers)). For positive integers n and m, define n WD the number of ways to divide a set of n elements into m m nonempty sets. ) 1 According to Knuth , Stirling (James, born in May, 1692 in Garden, Scotland—died on December 5, 1770 in Edinburgh, Scotland) first introduced the second kind. The names “first kind” and “second kind” are due to Nielsen (Niels, born on December 2, 1865 in Orslev, Denmark—died on September 16, 1931 in Copenhagen, Denmark), who first used these names in his book on the Gamma function [73, §26].
He presented these results in two papers16 submitted to Erlangen University on the occasion of becoming a member of the senate and the faculty. These results, however, caught no attention because, as Noether put it, von Staudt was so modest and did not care about the circulation of the papers (cf.  for the contents of these two papers). He also published a paper  on a proof of the “fundamental theorem on algebra” of Gauss (Clausen also wrote a paper on the same subject ). He wrote up a paper one week before his death, while suffering from asthma.
Bernoulli Numbers and Zeta Functions by Tsuneo Arakawa, Visit Amazon's Tomoyoshi Ibukiyama Page, search results, Learn about Author Central, Tomoyoshi Ibukiyama, , Masanobu Kaneko, Don B. Zagier