Number Theory

David Rohrlich

Course Description

We will start with the notion of a formal power series. While it is important that identities like

$\frac{1}{1-x} = \sum_{n\ge0}x^n \quad\quad (|x|<1)$

and

are equalities of real-valued functions, it is also important to be able to treat such equations as purely algebraic identities. Doing so allows one to prove, for example, that $e^{x+y}=e^xe^y$ , this identity being a simple consequence of the binomial theorem.

Next we will introduce the Bernoulli numbers $b_k$ . They are dened by the identity of formal power series

$\frac{t}{e^t-1}=\sum_{n\ge0}b_k\frac{t^k}{k!}$

We will also consider the Bernoulli polynomials $B_k(x)$ , which are dened by

$\frac{te^{xt}}{e^t-1}=\sum_{n\ge0}B_k(x)\frac{t^k}{k!}$

The Bernoulli numbers and Bernoulli polynomials have many remarkable proper- ties. For example, the nite sum

$1+2^k+3^k+\cdots + N^k$

has a simple expression in terms of Bernoulli polynomials, and the innite sum

$1+2^{-k}+3^{-k}+4^{-k}+\cdots$

has a simple expression in terms of Bernoulli numbers when $k$ is even. The latter sum is actually the value at $s=k$ of the Riemann zeta function $\zeta(s)$ , a function which is at the heart of the most famous unsolved problem in mathematics.

Number Theory

David Rohrlich

Course Description

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