Chebyshev's bias for products of irreducible polynomials

Abstract

For any $k\ge 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in ${\mathbf{F}}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. In the generic case, the bias dissipates as the degree of the modulus or $k$ gets large, but there are cases when the bias is extreme. 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. Several examples illustrate this new phenomenon.

The exposition has been improved, we now present the case of the number of irreducible factors both counting and not counting multiplicities. We also add some results on the possible values of the bias

References in corpus (9)

• Chebyshev's bias for analytic L-functions
• Vanishing of Hyperelliptic L-functions at the Central Point
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Cited by in corpus (7)

• Distribution of Frobenius elements in families of Galois extensions
• Roots of $L$-functions of characters over function fields, generic linear independence and biases
• Chebyshev's bias in dihedral and generalized quaternion Galois groups
• An annotated bibliography for comparative prime number theory
• Character sums over products of prime polynomials
• Sums of two squares are strongly biased towards quadratic residues
• Exceptional biases in counting primes over functions fields