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

2025/03/23 by Aymone, Marco, Chaves, Ana Paula, Ramos, Maria Eduarda
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.18228

Abstract

A modified Dirichlet character f is a completely multiplicative function such that for some Dirichlet character χ, f(p)=χ(p) for all but a finite number of primes p∈ S, and for those exceptional primes p∈ S, |f(p)|≤ 1. If χ is primitive and for each p∈ S we have |f(p)|=1, we prove that ∑_{n≤ x}f(n)=Ω((log x)^(|S|-3)/2). This makes progress on a Conjecture due to Klurman, Mangerel, Pohoata and Teräväinen, c.f. Trans. Amer. Math. Soc., 374 (2021), pp. 7967--7990. Our proof combines tools from Analytic Number Theory, Harmonic Analysis, Baker's Theory on linear forms in logarithms and Discrepancy bounds for sequences uniformly distributed modulo 1.

Related

• Ω-bounds for the partial sums of some modified Dirichlet characters
• Large character sums with multiplicative coefficients
• Large values of character sums with multiplicative coefficients
• Sharp bounds for maximal sums of odd order Dirichlet characters
• Large odd order character sums and improvements of the\n P 'olya-Vinogradov inequality