Representations of analytic functions as infinite products and their application to numerical computations

Let D be an open disk of radius ≤1 in $\mathbb{C}$ , and let (? _n ) be a sequence of ±1. We prove that for every analytic function $f: D \to \mathbb{C}$ without zeros in D, there exists a unique sequence (α _n ) of complex numbers such that $f(z) = f(0)\prod_{n=1}^{\infty} (1+\epsilon_{n}z^{n})^{\alpha_{n}}$ for every z∈D. From this representation we obtain a numerical method for calculating products of the form ∏ _p prime f(1/p) provided f(0)=1 and f′(0)=0; our method generalizes a well-known method of Pieter Moree. We illustrate this method on a constant of Ramanujan $\pi^{-1/2}\prod _{p~\text{prime}} \sqrt{p^{2}-p}\ln(p/(p-1))$ . From the properties of the exponents α _n , we obtain a proof of the following congruences, which have been the subject of several recent publications motivated by some questions of Arnold: for every n×n integral matrix A, every prime number p, and every positive integer k we have $\operatorname{tr} A^{p^{k}} \equiv\operatorname{tr} A^{p^{k-1}} { \hbox {\rm { (mod\ $p^{k}$) }}}$ .