Ω-bounds for the partial sums of some modified Dirichlet characters

Abstract

Abstract We consider the problem of Ω bounds for the partial sums of a modified character, i.e., a completely multiplicative function f such that $f(p)=\chi(p)$ for all but a finite number of primes p, where χ 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.