Approximation of Sums of Primes

TL;DR

This paper explores the approximation of slowly converging sums of primes using a reformulated prime number theorem and Riemann Zeta function values, providing error estimates for these approximations.

Contribution

It introduces a novel approach to approximate sums of primes with explicit error bounds based on the Riemann Zeta function.

Findings

Derived new approximation formulas for sums of primes

Provided explicit truncation error estimates

Validated the approach with numerical examples

Abstract

In this work we consider sums of primes that converging very slow. We set as a base, a reformulation of analytic prime number theorem and we use the values of Riemann Zeta function for the approximation. We also give the truncation error of these approximations