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Power series expansion connected with Riemann's zeta function Allard, Gabriel Louis Adolphe

Abstract

We consider the entire function [formula omitted] whose set of zeros includes the zeros of [formula omitted](s), expand it in an everywhere converging Maclauring series [formula omitted] Then we determine analytic expressions for the coefficients a[formula omitted] which will enable us to proceed with the numerical evaluation of some of these coefficients. To achieve this, we define an operator D[formula omitted] acting on a restricted class of power series and which we call the zeta operator. Using the operator D[formula omitted], we are able to express the coefficients a[formula omitted] as infinite n-dimensional integrals. Numerical values for the coefficients a₀ and a₁ are easily determined. For a₂ and a₃, we transform the multidimensional integrals into products of single integrals and obtain infinite series expressions for these coefficients. Although our method can also be used on the following coefficients, it turns out that the work involved to obtain an expression leading to a practical numerical evaluation of a₄, a₅, …,seems prohibitive at this stage. We then proceed with the numerical computation of a₂ and a₃ and we use these coefficients to calculate the sums of reciprocals of the zeros of [formula omitted](s) in the critical strip. Finally, assuming Riemann hypothesis, we calculate a few other quantities which may prove to be of interest.

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