Chebyshev's bias for products of irreducible polynomials

Lucile Devin, Xianchang Meng

Published 2018-09-25Version 1

For any $k\geq 1$, we compare the number of polynomials that have exactly $k$ irreducible factors in $\mathbf{F}_q[t]$ among different arithmetic progressions. We prove asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. We unconditionally derive the existence of the limiting distribution of this difference. In contrast to the case of products of $k$ prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. We give several examples to exhibit this new phenomenon.