The Erdős conjecture for primitive sets

Abstract

A subset of the integers larger than 1 is primitive if no member divides another. Erdős proved in 1935 that the sum of 1/(a log a) for a running over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on several fronts and show a connection to certain prime number "races" such as the race between π(x) and li(x).

Primes in prime number races

“…Moreover, we prove similar results-comparing the logarithmic density of a set of real numbers to the relative logarithmic density of the primes lying in that set-for a number of other prime races, some of which have not been considered before (see Theorems 4.1 and 5.6). These results resolve some problems from [12] and so make progress on the Erdős conjecture on primitive sets (see Section 4).…”

“…In [12] it was conjectured that all primes are Erdős-strong. Since 2 is not a Mertens prime, it would be great progress just to be able to prove that 2 is Erdős-strong.…”

Almost primes and the Banks–Martin conjecture

“…Zhang [13] proved that f (N 1 ) > f (N k ) for each k ≥ 2. Lichtman and Pomerance [7] proved that f (N k ) ≫ 1. Bayless, Kinlaw, and Klyve [2] recently showed that f (N 2 ) > f (N 3 ), providing bounds on f (N k ), f (N * k ) for small k. Here N * k is the set of squarefree k-almost primes.…”

The degree of primitive sequences and Erdős conjecture

“…In [4], Erdős and Zhang showed that for any primitive sequence , and in [5], Clark improved this result (where γ is the Euler constant) in the special case when A is a primitive set of composite numbers. Several years later in [6],…”