Remarks on a Formula of Ramanujan

TL;DR

This paper investigates the detailed structure of Ramanujan's formula under specific assumptions related to the Riemann zeta function and Mertens' conjecture, offering insights into number theory.

Contribution

It introduces a novel analysis of Ramanujan's formula assuming averaged Mertens' conjecture and linear independence of zeta zeros' ordinates.

Findings

Enhanced understanding of Ramanujan's formula structure

Implications for the distribution of zeta zeros

Connections to Mertens' conjecture

Abstract

Assuming an averaged form of Mertens' conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyze the finer structure of the terms in a well-known formula of Ramanujan.