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Let  be an open disk of radius ≤1 in , and let ( ) be a sequence of ±1. We prove that for every analytic function  without zeros in , there exists a unique sequence ( ) of complex numbers such that  for every ∈. From this representation we obtain a numerical method for calculating products of the form ∏ (1/) provided (0)=1 and ′(0)=0; our method generalizes a well-known method of Pieter Moree. We illustrate this method on a constant of Ramanujan . From the properties of the exponents  , 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 × integral matrix , every prime number , and every positive integer  we have .

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