$\Omega$-bounds for the partial sums of some modified Dirichlet characters

TL;DR

This paper establishes $ ext{Omega}$ bounds for partial sums of modified Dirichlet characters, showing they grow at least as fast as a power of log x in certain cases, and explores related Diophantine properties.

Contribution

It proves $ ext{Omega}$ bounds for partial sums of modified Dirichlet characters and connects these bounds to Diophantine properties of logarithms of primes, refining a conjecture by Klurman et al.

Findings

Partial sums grow at least as fast as $( ext{log } x)^{|S|}$ in special cases

Computed Riesz means for large order $k$ of modified characters

Linked Diophantine properties of $rac{ ext{log } p}{ ext{log } q$ to average behaviors

Abstract

We consider the problem of $\Omega$ bounds for the partial sums of a modified character, \textit{i.e.}, a completely multiplicative function $f$ such that $f(p)=\chi(p)$ for all but a finite number of primes $p$, where $\chi$ is a primitive Dirichlet character. We prove that in some special circumstances, $\sum_{n\leq x}f(n)=\Omega((\log x)^{|S|})$, where $S$ is the set of primes $p$ where $f(p)\neq \chi(p)$. This gives credence to a corrected version of a conjecture of Klurman et al., Trans. Amer. Math. Soc., 374 (11), 2021, 7967-7990. We also compute the Riesz mean of order $k$ for large $k$ of a modified character, and show that the Diophantine properties of the irrational numbers of the form $\log p / \log q$, for primes $p$ and $q$, give information on these averages.