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

The article investigates Ω bounds for the partial sums of modified characters, defined as completely multiplicative functions f that differ from a primitive Dirichlet character χ at only finitely many primes. It establishes that under certain conditions—specifically when χ is primitive and the modifications at primes have absolute value one—the partial sums satisfy the lower bound \(\sum_{n \leq x} f(n) = \Omega((\log x)^{|S|})\), where S is the finite set of primes where f differs from χ. The paper further computes the Riesz means of order k for such modified characters, showing that the exponent in the Ω bound is optimal in this averaged context, and connects the analytic behavior of these means to Diophantine properties of numbers of the form \(\log p / \log q\) for primes p and q. These results refine and partially confirm a conjecture by Klurman et al. regarding the growth of partial sums of modified characters.