Halász’s theorem for Beurling numbers
Volume 183 / 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.
