Twenty Digits of Some Integrals of the Prime Zeta Function

TL;DR

This paper accurately computes a complex double sum involving primes and their powers to 24 decimal places, using numerical methods based on the Prime Zeta Function's properties.

Contribution

It introduces a precise numerical evaluation of a prime-related double sum using a method adapted from Cohen's approach to the Prime Zeta Function.

Findings

Sum value approximated to 24 decimal digits

Method based on Prime Zeta Function derivatives

Numerical strategy effectively evaluates complex sums

Abstract

The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohen's work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.