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

Abstract

Let $D$ be an open disk of radius $\le 1$ in $\mathbb{C}$, and let $({\epsilon }_n)$ be a sequence of $\pm 1$. We prove that for every analytic function $f:D\to \mathbb{C}$ without zeros in $D$, there exists a unique sequence $({\alpha }_n)$ of complex numbers such that $f(z)=f(0){\prod }_{n=1}^{\infty }(1+{\epsilon }_n{z}^n{)}^{{\alpha }_n}$ for every $z\in D$. From this representation we obtain a numerical method for calculating products of the form ${\prod }_{p\text{prime}}f(1/p)$ provided $f(0)=1$ and $f^{\prime }(0)=0$; 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 ${\alpha }_n$, 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 $n\times n$ integral matrix $A$, every prime number $p$, and every positive integer $k$ we have $\text{tr}{A}^{p^k}\equiv \text{tr}{A}^{p^{k-1}}(\text{mod}{p}^k)$.

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