# MUIC Math

Mathematics at MUIC

### Internal

seminar:at_bernoulli

# Arithmetic Properties of the Bernoulli Numbers

Aram Tangboonduangjit
Mahidol University International College, Nakhon Pathom

16:00 Friday, 19 May 2017, Room 3303, MUIC

### Abstract

The Bernoulli numbers $B_n$ appear as coefficients in a Maclaurin series expansion of the function $x\mapsto x/(e^x-1).$ Some of their number-theoretic properties have a connection with the Euler-Maclaurin-sum formula: For $k\ge 0$ and $n\ge 2,$ $$1^k + 2^k + \cdots + (n-1)^k = \sum_{r=0}^k \frac{1}{k+1-r}{k\choose r}n^{k+1-r}B_r.$$ Their fractional parts are realized by a celebrated theorem of von Staudt and Clausen (1840) which states the following: For $k\ge 1,$ $$B_{2k}+\sum_{\substack{p\, \mathrm{prime} \\ p-1 \mid 2k}}\frac{1}{p} \in \mathbb{Z}.$$ In this talk, we will prove the above two results and discuss some recent developments of arithmetic properties of the Bernoulli numbers.