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## Acta Arithmetica

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## Halász’s theorem for Beurling numbers

### Volume 183 / 2018

Acta Arithmetica 183 (2018), 223-235 MSC: Primary 11N37; Secondary 11N05. DOI: 10.4064/aa8668-11-2017 Published online: 27 April 2018

#### Abstract

Halász’s mean-value theorem is well-known and important in classical probabilistic number theory. It is generalized to Beurling generalized numbers as follows.

Let $f(n_i)$ be a completely multiplicative function on Beurling generalized numbers $\mathcal{N}$ such that $|f(n_i)|\le 1$ for all $n_i\in \mathcal{N}$. Suppose (1) $N(x) \sim Ax$ and $\int_1^\infty x^{-\sigma-1}|N(x)-Ax|\,dx= O((\sigma-1)^{-\beta})),\quad \sigma\to 1+,$ with some constants $A \gt 0$ and $\beta\in [0,\, 1/2)$ and (2) the Chebyshev function $\psi(x)$ satisfies $\psi(x)\ll x$. If the Halász condition $\hat F(s):=\sum_{i=1}^\infty \frac{f(n_i)}{n_i^s}=\frac{c}{s-1}+o\biggl(\frac{1}{\sigma-1}\biggr)$ holds as $\Re s=\sigma\to 1+$ uniformly for $-K\le t\le K$ ($t=\mathfrak Is$) for each fixed $K \gt 0$ then $F(x):=\sum_{n_i\le x}1=cx +o(x).$

This implies further a generalization of the Halász–Wirsing mean-value theorem for Beurling numbers $\mathcal{N}$ with the same conditions on $N(x)$ and $\psi(x)$. It follows that $\psi(x)\ll x$ implies the estimate $M(x)=o(x)$ for $\mathcal{N}$. In contrast to classical number theory, one conjectures that the two estimates are equivalent in $\mathcal{N}$. However, whether $M(x)=o(x)$ implies $\psi(x)\ll x$ is undetermined.

#### Authors

• Wen-Bin ZhangDepartment of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA
e-mail

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