2012/02/06 by Marcin Mazur, Mazur, Marcin, Bogdan V. Petrenko +1
Mathematics · #Mathematical and Theoretical Analysis #math.NT #msc:11A07 #msc:11C20 #msc:11Y60 #msc:30E10 #msc:30J99 #msc:40A20 #msc:40A30
paper · pdf · doi:10.48550/arxiv.1202.1335
Several editorial changes have been made
arxiv created 2012/04/04 · arxiv updated 2012/04/05
Let D be an open disk of radius ≤ 1 in \mathbb C, and let (εn) be a sequence of ± 1. We prove that for every analytic function f: D → \mathbb C without zeros in D, there exists a unique sequence (αn) of complex numbers such that f(z) = f(0)∏n=1∞ (1+εnzn)αn 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 π-1/2∏p prime √(p2-p)ln(p/(p-1)). From the properties of the exponents αn, 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 tr Apk ≡ tr A^pk-1 (mod pk).