Exceptional biases in counting primes over function fields

Abstract

AbstractWe 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 as , a power of a fixed prime, goes to infinity. The bounds given extend a previous result of Kowalski, who studied a similar question along particular one‐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.