Work

Weighted Prime Number Theorem on Arithmetic Progressions with Refinements

Year: 2025

Type: article

Abstract: We extend Dirichlet’s prime number theorem on arithmetic progressions to the weighted counting function πs(x; q, a) for (a, q) = 1 with the weight p−w for prime p with w ≥ 0. We also prove that when 0 ≤ w < 1/2, the difference πw(x; q, a) − πw(x; q, b) changes its sign infinitely many times as x grows for any coprime a, b (a ̸= b) with p under the assumption that Dirichlet L-functions have no real nontrivial zeros.. This result gives a justification of the theory of Aoki-Koyama that Chebyshev’s bias is formulated by the asymptotic behavior of πw(x; q, a) − πw(x; q, b) at s = 1/2.

Source: Mathematics

DOI: 10.3390/math13162564

Authors: Shin-ya Koyama, Koji Shimada

Institution: Toyo University

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: Toyo University