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

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.