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

2022/10/12 by Marco Aymone, Aymone, Marco
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2210.06153

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)=χ(p) for all but a finite number of primes p, where χ is a primitive Dirichlet character. We prove that in some special circumstances, ∑_{n≤ x}f(n)=Ω((log x)^|S|), where S is the set of primes p where f(p)≠ χ(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.

Related

• Ω-bounds for the partial sums of some modified Dirichlet characters II
• Large values of character sums with multiplicative coefficients
• Large character sums with multiplicative coefficients
• Modified Dirichlet character sums over the k-free integers
• Character sums over shifted primes