Modeling Chebyshev's Bias in the Gaussian Primes as a Random Walk

TL;DR

This paper models Chebyshev's bias in Gaussian primes as a Legendre symbol random walk, revealing correlations and extending the analysis to number fields, providing visual insights into prime residue biases.

Contribution

It introduces a novel Legendre symbol walk model for Chebyshev's bias, including extensions to Gaussian primes and analysis of correlations between different prime norms.

Findings

Chebyshev's bias may be reduced by restricting to nonquadratic residues.

Strong correlations observed between Legendre symbol walks for Gaussian primes with equal norms.

Explanations proposed for positive and negative correlations based on prime norms.

Abstract

One aspect of Chebyshev's bias is the phenomenon that a prime number, $ q $, modulo another prime number, $ p$, experimentally seems to be slightly more likely to be a nonquadratic residue than a quadratic residue. We thought it would be interesting to model this residue bias as a "random" walk using Legendre symbol values as steps. Such a model would allow us to easily visualize the bias. In addition, we would be able to extend our model to other number fields. In this report, we first outline underlying theory and some motivations for our research. In the second section, we present our findings in the rational prime numbers. We found evidence that Chebyshev's bias, if modeled as a Legendre symbol $ \left(\frac{q}{p}\right)$ walk, may be somewhat reduced by only allowing $ q$ to iterate over primes with nonquadratic residue (mod $ 4$). In the final section, we extend our Legendre symbol walks to the Gaussian primes and present our main findings. Let $ \pi_1 = \alpha+\beta i$ and $ \pi_2 = \beta+\alpha i$. We observed strong ($ \pm$) correlations between Gaussian Legendre symbol walks for $ \left[\frac{a+bi}{\pi_1}\right]$ and $ \left[\frac{a+bi}{\pi_2}\right]$ where $ N(\pi_1) = N(\pi_2)$ and $ a+bi$ iterates over Gaussian primes in the first quadrant. We attempt an explanation of why, for some norms, the plots for $ \pi_1$ and $ \pi_2$ have strong positive correlation, while, for other norms, the plots have strong negative correlation. We hope to have written in a way that makes our observations accessible to readers without prior formal training in number theory.