Remainder in Modified Mertens Formula and Ramanujan Inequality

TL;DR

This paper establishes a strong upper estimate in the modified Mertens formula as both necessary and sufficient for the validity of the Ramanujan inequality, providing new criteria related to prime reciprocals.

Contribution

It introduces a new necessary and sufficient upper estimate in the modified Mertens formula for prime reciprocals linked to Ramanujan inequality.

Findings

Proves the necessity and sufficiency of a strong upper estimate in the modified Mertens formula

Derives additional criteria related to prime reciprocals and inequalities

Connects asymptotic estimates with Ramanujan inequality validity

Abstract

A highly strong upper estimate in the modified asymptotic formula for sums of the primes' reciprocals is proved to be necessary (as well as sufficient) in order the Ramanujan inequality holds true. Some other criteria in similar terms are also obtained.