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Chebyshev's bias for irrational factor function

Bittu Chahal

TL;DR

This work investigates the distribution of the irrational factor function $I_k(n)$ and its natural analogues over $\mathbb{Q}$, number fields, and function fields, focusing on summatory functions and Chebyshev’s bias phenomena. It develops asymptotic formulas for the corresponding sums $S_{k,\mathbb{K}}(x)$ and $\mathfrak{S}_k(N; \mathfrak{m},\mathfrak{g})$ via Wiener–Ikehara Tauberian theory and contour integration, with main terms expressed through residues of zeta and $L$-functions (e.g., $\lambda_{\mathbb{K}}$, $R_k(2)$, $\mathcal{L}(k,\chi_0)$). The paper also establishes $\Omega$-results for error terms and demonstrates Littlewood-type sign changes, revealing Chebyshev-type bias in arithmetic progressions modulo ideals or polynomials, under standard hypotheses such as GLH and Haselgrove-type zero-free regions (or their function-field analogues where RH holds). These results illuminate bias phenomena across both classical and function-field settings and extend the scope of irrational-factor investigations beyond the integers to algebraic settings. The methods fuse Dirichlet/Hecke $L$-functions, Tauberian theorems, and zero-distribution assumptions to quantify the size and sign oscillations of the relevant summatory functions, with potential implications for prime-divisor races in generalized arithmetic settings.

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.

Chebyshev's bias for irrational factor function

TL;DR

This work investigates the distribution of the irrational factor function and its natural analogues over , number fields, and function fields, focusing on summatory functions and Chebyshev’s bias phenomena. It develops asymptotic formulas for the corresponding sums and via Wiener–Ikehara Tauberian theory and contour integration, with main terms expressed through residues of zeta and -functions (e.g., , , ). The paper also establishes -results for error terms and demonstrates Littlewood-type sign changes, revealing Chebyshev-type bias in arithmetic progressions modulo ideals or polynomials, under standard hypotheses such as GLH and Haselgrove-type zero-free regions (or their function-field analogues where RH holds). These results illuminate bias phenomena across both classical and function-field settings and extend the scope of irrational-factor investigations beyond the integers to algebraic settings. The methods fuse Dirichlet/Hecke -functions, Tauberian theorems, and zero-distribution assumptions to quantify the size and sign oscillations of the relevant summatory functions, with potential implications for prime-divisor races in generalized arithmetic settings.

Abstract

In this article, we study the distribution of the irrational factor function of order , introduced first by Atanassov for and later it was generalized by Dong et al. for all . 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.
Paper Structure (20 sections, 18 theorems, 109 equations)

This paper contains 20 sections, 18 theorems, 109 equations.

Table of Contents

  1. Introduction and main results
  2. Irrational factor function over $\mathbb{Q}$
  3. Irrational factor function over number fields
  4. Irrational factor function over function fields
  5. Notation
  6. Acknowledgements
  7. Preliminaries
  8. Bounds for the Riemann zeta function and Dirichlet $L$-function modulo $q$
  9. Bounds for Dedekind zeta function and Hecke $L$-function with GLH
  10. Bounds for Riemann zeta function and Dirichlet $L$-function over function fields
  11. Number fields
  12. Proof of Theorem \ref{['thm6 Tauberian']}
  13. Proof of Theorem \ref{['thm4']}
  14. Proof of Theorem \ref{['thm6 Tauberian2']}
  15. Proof of Theorem \ref{['thm5']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

For $k\geqslant 2$, we have where $\lambda_{\mathbb{K}}$ is the residue of the Dedekind zeta function at $s=1$ and ${R}_{k}(2)$ is a constant depending on $k$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 13 more