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

TL;DR

This paper introduces a novel representation of zero-free analytic functions as infinite products with unique exponents, enabling a new numerical method for prime product calculations and proving related congruences.

Contribution

It establishes a unique infinite product representation for zero-free analytic functions and applies it to develop a numerical method for prime products, extending previous techniques.

Findings

Derived a unique product representation for analytic functions without zeros.

Developed a numerical method for calculating prime products using this representation.

Proved congruences related to traces of powers of integer matrices modulo prime powers.

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 $f(z) = f(0)\prod_{n=1}^{\infty} (1+\epsilon_nz^n)^{\alpha_n}$ 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 $f'(0) = 0$; 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 $\text{tr} A^{p^k} \equiv \text{tr} A^{p^{k-1}} (\text{mod}\,{p^k})$.