A Strengthening of Theorems of Halász and Wirsing

Abstract

Given an arithmetic function $g(n)$ write $M_g(x):={\sum }_{n\le x}g(n)$. We extend and strengthen the results of a fundamental paper of Halász in several ways by proving upper bounds for the ratio of $\frac{|M_g(x)|}{M_{|g|}(x)}$, for any strongly multiplicative, complex-valued function $g(n)$ under certain assumptions on the sequence $\{g(p){\}}_p$. We further prove an asymptotic formula for this ratio in the case that $|\text{arg}(g(p))|$ is sufficiently small uniformly in $p$. In so doing, we recover a new proof of an explicit lower mean value estimate for $M_f(x)$ for any non-negative, multiplicative function satisfying $c_1\le |f(p)|\le c_2$ for $c_2\ge c_1>0$, by relating it to $\frac{x}{\log x}{\prod }_{p\le x}\left(1+\frac{f(p)}{p}\right)$. As an application, we generalize our main theorem in such a way as to give explicit estimates for the ratio $\frac{|M_g(x)|}{M_f(x)}$, whenever $f:\mathbb{N}\to (0,\infty )$ and $g:\mathbb{N}\to \mathbb{C}$ are strongly multiplicative functions that are uniformly bounded on primes and satisfy $|g(n)|\le f(n)$ for every $n\in \mathbb{N}$. This generalizes a theorem of Wirsing and extends recent work due to Elliott.

43 pages; corrections to main result and proofs, along with some extensions

References in corpus (3)

• Large values of $L(1,\chi )$ for $k$-th order characters $\chi$ and applications to character sums
• On the Number of Restricted Prime Factors of an Integer II
• On the Distribution of Integers with Restricted Prime Factors I

Cited by in corpus (2)

• Linear correlations of multiplicative functions
• On the Number of Restricted Prime Factors of an Integer II