Chebyshev's bias for products of irreducible polynomials
| Autor: | Devin, Lucile, Meng, Xianchang |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Advances in Mathematics Volume 392, 3 December 2021, 108040 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.aim.2021.108040 |
| Popis: | For any $k\geq 1$, 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. Comment: 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 |
| Databáze: | arXiv |
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