Halász's theorem for Beurling generalized numbers

Abstract

We show that Halász's theorem holds for Beurling numbers under the following two mild hypotheses on the generalized number system: existence of a positive density for the generalized integers and a Chebyshev upper bound for the generalized primes.∞ 1 − x −s dG(x) and we use the notation s = σ + it for complex variables. Zhang's version of Halász's theorem reads as follows. His result generalizes [5, Theorem 3.1], where the set of hypotheses (1.1), (1.2), and (1.3) were actually introduced.Theorem 1.1 (Zhang [17]). Suppose that the generalized number system satisfies a Chebyshev upper estimate

The average order of the Möbius function for Beurling primes

“…Zhang [10] was the first to note that PNT is not equivalent to M P (x) = o(x). For the most general results giving M P (x) = o(x), see the very recent papers [3] and [4].…”

Completely multiplicative functions with sum zero over generalised prime systems

“…A function f : N −→ C is said to be multiplicative on N if f (1) = 1 and f (mn) = f (m)f (n) whenever (m, n) = 1. Such an f is said to be completely multiplicative [5,16] if we also have f (mn) = f (m)f (n) for all values of m, n ∈ N , where (m, n) is defined as the largest ginteger that divides both m and n. We define the generalised Liouville function on N , is an example of completely multiplicative function, to be λ P (1) = 1 and λ P (n) = (−1)…”

Beurling numbers whose number of prime factors lies in a specified residue class

“…Hence, dF q = exp * (h q dΠ), as the Mellin-Stieltjes transform is an injective operation. We now wish to apply Theorem 2.3 from [3]. We split h q into g 1 + g 2 as e 2πiq/K + (h q − e 2πiq/K ).…”

“…This topic was introduced by Zhang in [5] whose results he later improved upon in [6] based on ideas of [2]. The paper [3] refines these ideas further and contains the best results currently available.…”

“…The main goal of this paper is to prove the following theorem. A crucial ingredient of the proof is a recently established version [3] of Halász's theorem for Beurling numbers. This topic was introduced by Zhang in [5] whose results he later improved upon in [6] based on ideas of [2].…”