Chebyshev's bias for irrational factor function

TL;DR

This paper investigates the distribution and bias phenomena of the irrational factor function across number fields and function fields, providing asymptotic formulas and error estimates for its average behavior.

Contribution

It introduces the irrational factor function in new settings and derives asymptotic and omega results, extending previous work on its distribution and bias.

Findings

Derived asymptotic formulas for average values

Established omega results for error terms

Analyzed Chebyshev's bias in new contexts

Abstract

In this article, we study the distribution of the irrational factor function of order $k$, introduced first by Atanassov for $k=2$ and later it was generalized by Dong et al. for all $k\geq 2$. We introduce the irrational factor function in both number field and function field settings, derive asymptotic formulas for their average value, and further establish omega results for the error term in the asymptotic formulas. Moreover, we study the Chebyshev's bias phenomenon for number field and function field analogues of sum of the irrational factor function.