On certain sums over primes and the Riesz function

TL;DR

This paper explores series involving the Möbius function that approximate prime sums, introducing a new formula linking the Gram series derivative to the Riesz function using Riemann-Stieltjes integration.

Contribution

It presents a novel formula connecting the Gram series derivative to the Riesz function, enabling new integral relationships for prime sum approximations.

Findings

Derived a new formula linking Gram series derivative and Riesz function

Established general integral relationships involving prime sums

Utilized Riemann-Stieltjes integration in the analysis

Abstract

We offer some comments on series involving the M$\ddot{o}$bius function which approximate sums over primes. To accomplish this, we utilize the derivative of the Gram series by applying Riemann-Stieltjes integration. We offer a new formula connecting the derivative of the Gram series to the Riesz function, which we then use to obtain general integral relationships.