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Thursday, March 8, 2018

**Abstract:** Chebyshev observed that there seems to be more primes congruent to 3 mod 4 than those congruent to 1 mod 4, which is known as Chebyshev’s bias. In this talk, we introduce two generalizations of this phenomenon. 1) Greg Martin conjectured that the difference of the summatory function of the number of prime factors over integers less than x from different arithmetic progressions will attain a constant sign for sufficiently large x. Under some reasonable conjectures, we give strong evidence to support this conjecture. 2) We introduce the function field version of Chebyshev’s bias. We consider the distribution of products of irreducible polynomials over finite fields. When we compare the number of such polynomials among different arithmetic progressions, new phenomenon will appear due to the existence of real zeros for some associated L-functions.