Chebyshev's bias for irrational factor function

2:00pm, Wednesday 1 Oct 2025

Abstract

Chebyshev’s bias is the phenomenon that the number of prime quadratic nonresidues of a given modulus predominate over the prime quadratic residues, in other words, primes are biased toward quadratic nonresidues. We study this bias question in the context of the irrational factor function $I_k(n)$, defined by $I_k(n) = \Pi_{i=1}^l p_i^{\beta_i}$, where $n=\Pi_{i=1}^l p_i^{\beta_i}$ and $\beta_i = \alpha_i$ if $\alpha_i <k $ and $\beta_i = 1/\alpha_i$ if $\alpha_i \ge k$.

In particular, we introduce the irrational factor function in both number field and function field settings, derive asymptotic formulas for their average value, and establish Ω-results for the error term in the asymptotic formulas. Furthermore, we study the Chebyshev’s bias phenomenon for number field and function field analogues of sum of the irrational factor function, under assumptions on the real zeros of Hecke L-functions associated with Hecke characters in the number field case.