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Mathematics, Number Theory, math.NT,Mathematics, Combinatorics
Abstract:
We introduce a simple sieve-theoretic approach to studying partial sums of
multiplicative functions which are close to their mean value. This enables us
to obtain various new results as well as strengthen existing results with new
proofs.
As a first application, we show that for a completely multiplicative function
$f : \mathbb{N} \to \{-1,1\},$ \begin{align*}
\limsup_{x\to\infty}\Big|\sum_{n\leq x}\mu^2(n)f(n)\Big|=\infty. \end{align*}
This confirms a conjecture of Aymone concerning the discrepancy of square-free
supported multiplicative functions.
Secondly, we show that a completely multiplicative function $f : \mathbb{N}
\to \mathbb{C}$ satisfies \begin{align*} \sum_{n\leq x}f(n)=cx+O(1)
\end{align*} with $c\neq 0$ if and only if $f(p)=1$ for all but finitely many
primes and $|f(p)|<1$ for the remaining primes. This answers a question of
Ruzsa.
For the case $c = 0,$ we show, under the additional hypothesis $$\sum_{p
}\frac{1-|f(p)|}{p} < \infty,$$ that $f$ has bounded partial sums if and only
if $f(p) = \chi(p)p^{it}$ for some non-principal Dirichlet character $\chi$
modulo $q$ and $t \in \mathbb{R}$ except on a finite set of primes that
contains the primes dividing $q$, wherein $|f(p)| < 1.$ This provides progress
on another problem of Ruzsa and gives a new and simpler proof of a stronger
form of Chudakov's conjecture.
Along the way we obtain quantitative bounds for the discrepancy of the
generalized characters improving on the previous work of Borwein, Choi and
Coons.