Exceptional biases in counting primes over functions fields

Abstract

We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias", "lower order bias" and "reversed bias", occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in ${\mathbb{F}}_q[x]$ as $q$, a power of a fixed prime, goes to infinity. The bounds given improve on a previous result of Kowalski, who studied a similar question along particular $1$-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret-Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

arXiv comment22 pages. Comments welcome

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• Vanishing of Hyperelliptic L-functions at the Central Point
• The large sieve, monodromy and zeta functions of algebraic curves, II: independence of the zeros
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Cited by in corpus (1)

• An annotated bibliography for comparative prime number theory