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Journal Article

#### Multiplicative functions that are close to their mean

##### External Resource

https://doi.org/10.1090/tran/8427

(Publisher version)

https://doi.org/10.48550/arXiv.1911.06265

(Preprint)

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##### Citation

Klurman, O., Mangerel, A. P., Pohoata, C., & Teräväinen, J. (2021). Multiplicative
functions that are close to their mean.* Transactions of the American Mathematical Society,*
*374*(11), 7967-7990. doi:10.1090/tran/8427.

Cite as: https://hdl.handle.net/21.11116/0000-0009-903A-3

##### 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.

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.