A PNT equivalence for Beurling numbers

Abstract

In classical prime number theory, several relations are considered to be equivalent to the Prime Number Theorem. For Beurling generalized numbers, some auxiliary conditions may be needed to deduce one relation from another one. We show conditions under which the Beurling analog of the sharp version of Mertens' sum formula does or does not hold.

On PNT equivalences for Beurling numbers

“…if α > 1. Interestingly, (1.8) with α = 1 and the PNT are not strong enough to ensure the sharp Mertens relation, as established by an example in [11,12]. We will strengthen that result as well.…”

“…which, as in [11], we call a sharp Mertens relation. Note that for the ordinary rational primes (1.2) holds with c = −γ, where γ is the Euler-Mascheroni constant; however, in general, we may have c = −γ.…”

Prime Number Theorem Equivalences and Non‐equivalences

“…Here * denotes multiplicative convolution. The right side of (3.3) equals 4) and the left side, after integration by parts, is…”

“…We can establish (2.1) using a slightly weaker integral hypothesis, but, as the following example from [4] shows, for estimates of the type (3.2), γ > 1 is optimal: take…”

Explicit Estimates: From Λ(n) in Arithmetic Progressions to Λ(n)/n

“…[7] indicates that this inequality does not hold in the case of Beurling generalized integers without any further assumption; interestingly, [21] shows that an equivalent of the Mertens formula is always valid in any Beurling system. Here is a theorem that quantifies the strength of our approach: Theorem 1.1.…”