Studying the secondary terms of the Prime Number Theorem in Arithmetic Progressions, Chebyshev claimed that there are more prime numbers congruent to 3 modulo 4 than to 1 modulo 4. This claim was later corrected by Littlewood, explained, and quantified by Rubinstein and Sarnak.
Pursuing the work of Cha, we investigate analogues to Chebyshev's bias in the setting of irreducible polynomials over finite fields. In particular, we observe exceptional behaviors occurring when the zeros of the involved L-functions are not linearly independent. More precisely, we will present instances of "complete bias" and "reversed bias", and explain why they occur with probability tending to 0, in the large q limit.

This is joint work with Bailleul, Keliher and Li