Representations of analytic functions as infinite products and their application to numerical computations
type: Item
instance of: scholarly article
title: Representations of analytic functions as infinite products and their application to numerical computations
zbMATH Open document ID: 1317.30055
author: Marcin Mazur
author: Bogdan V. Petrenko
published in: The Ramanujan Journal
publication date: 2014-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.1335
review text: Let \(D\) be an open disk of radius smaller than or equal to 1 in the complex plane and let \(f\) be an analytic function from \(D\) to the complex numbers without zeros in \(D\) and further satisfying \(f(0)=1\) and \(f'(0)=0\). Many constants arising in number theory, especially in analytic number theory, are of the form \(\prod_{p}f(1/p)\) (they usual arise in density problems). The authors present a method to compute such constants with very high numerical accuracy by representing the product as an infinite product involving zeta values evaluated at integers \(\geq 2\). This generalizes work of the reviewer [Manuscr. Math. 101, No. 3, 385--399 (2000; Zbl 1007.11084)], who only considered the case where \(f\) is rational. The starting point of the present investigation is to show that given a sequence of \((\epsilon_n)\) with \(\epsilon_n\in \{\pm 1\}\), there exists a unique sequence \((\alpha_n)\) of complex numbers such that \(f(z)=f(0)\prod_{n=1}^{\infty}(1+\epsilon_nz^n)^{\alpha_n}\). The authors demonstrate their method on a function \(f\) that is not rational and is closely related to a constant in the work of Ramanujan. As a by-product they obtain an elegant reproof of the result (conjectured by V. Arnold) that given an \(n \times n\) integral matrix \(A\) and a prime number \(p\) the trace of \(A^{p^k}\) is congruent to the trace of \(A^{p^{k-1}}\) modulo \(p^k\).
reviewed by: Pieter Moree
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 11Y60
Mathematics Subject Classification ID: 30J99
Mathematics Subject Classification ID: 40A30
Mathematics Subject Classification ID: 40A20
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11C20
zbMATH DE Number: 6348378
zbMATH Keywords: Euler product
zbMATH Keywords: infinite products
zbMATH Keywords: approximating constants
zbMATH Keywords: congruences for traces
MaRDI profile type: Publication
OpenAlex ID: W2593997777
arXiv ID: 1202.1335
cites work: Q3724706
cites work: Q4218535
cites work: The matrix Euler-Fermat theorem
cites work: On the matricial version of Fermat-Euler congruences
cites work: Q3066834
cites work: Q3934484
cites work: On the analytic continuation of Eulerian products
cites work: Q4936071
cites work: Generalizations of Arnold's version of Euler's theorem for matrices
cites work: Approximation of singular series and automata
cites work: The product over all primes is \(4\pi^{2}\)
cites work: Q5766728
cites work: On some number-theoretic conjectures of V. Arnold
cites work: On matrix analogs of Fermat's little theorem
cites work: On congruences for the traces of powers of some matrices
DOI: 10.1007/S11139-013-9546-3
Recommended article: On the Freund-Witten adelic formula for Veneziano amplitudes
Recommended article: A Convergent Series for the Partition Function p ( n )
Recommended article: Q4695880
Recommended article: Note on the subject of the communication of M. Stieltjes ''On a uniform function''.
Recommended article: On the approximation of certain infinite products.
Recommended article: Sobre los ceros de las funciones ciclicas.
Recommended article: GLAISHER-TYPE PRODUCTS OVER THE PRIMES
Recommended article: On some formulae in the theory of the zeta-function.
Recommended article: Q3073840
Recommended article: Q3676221
description: scientific article; zbMATH DE number 6348378