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

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. [ABSTRACT FROM AUTHOR]