Representations of analytic functions as infinite products and their application to numerical computations
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 \(p^k\).
scientific article; zbMATH DE number 6348378
scientific article