On Maximal Values of Gronwall Numbers for Integers with Given Greatest Prime Factor and Remainder in Modified Mertens Formula

TL;DR

This paper establishes an unconditional limit relationship between the remainder in a modified Mertens formula and the maximal Gronwall numbers for integers with a fixed greatest prime factor, advancing understanding without assuming the Riemann Hypothesis.

Contribution

It provides the first unconditional limit relationship linking the Mertens formula remainder to maximal Gronwall numbers for specific integers.

Findings

Unconditional limit relationship derived

Maximal Gronwall numbers characterized for integers with prime factor p

Advances understanding of prime reciprocals sum asymptotics

Abstract

The unconditional, i.e. without assuming validity of RH, sharp limit relationship (as p tends to infinity) is found between the remainder in the modified Mertens asymptotic formula for the sums of primes' reciprocals and maximal values of Gronwall numbers G(N) among all integers whose greatest prime factor is p and which are divided by any prime q<p. The proofs are based on the properties of G(N) studied in previous author's preprints.