The average order of the Möbius function for Beurling primes

Abstract

In this paper, we study the counting functions [Formula: see text], [Formula: see text] and [Formula: see text] of a generalized prime system [Formula: see text]. Here, [Formula: see text] is the partial sum of the Möbius function over [Formula: see text] not exceeding [Formula: see text]. In particular, we study these when they are asymptotically well-behaved, in the sense that [Formula: see text], [Formula: see text] and [Formula: see text], for some [Formula: see text] and [Formula: see text]. We show that the two largest of [Formula: see text] must be equal and at least [Formula: see text].