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Reaching every possible Chebyshev's bias in function fields

Posted in
Lucile Devin
University of Montreal
Wed, 2019-09-25 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

For every given integer m, Chebyshev's original interrogation is to investigate on the probability ("the bias") that given a random real number x, there are more prime numbers lower than x that are non-quadratic residues than that are quadratic residues modulo m.
In a joint work with X. Meng, we investigate on the translation of this question in function fields. This translation was initiated by Cha, who observed biases in "unexpected directions" when there are linear relations between the zeros of the L-functions. We show that almost anything can happen depending on the modulus m, in particular one can have complete biases (=1), no bias at all (=1/2), and, conditionally on a hypothesis of linear independence of the zeros, the bias can approach any value in the interval [1/2,1].

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