Web Page of the Ottawa-Carleton Number Theory Seminar

Year 2017-2018

Abstract: Following ideas of Rubinstein, Sarnak and Fiorilli, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions L(s) = ∑ _n≥1λ_f(n) n^-s satisfying natural analytic hypotheses. We put the emphasis on weakening the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular we establish the existence of the logarithmic density of the set {x ≥ 2 : ∑ _p≤xλ_f(p) ≥ 0} conditionally on a much weaker hypothesis than was previously known. We include applications to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases.

Title: Large gaps in the values of positive-definite cubic and biquadratic diagonal forms

Abstract: Let S be the set of the natural numbers that can be written as a sum of 3 nonnegative cubes, or more generally are the values of a given (homogeneous, positive-definite, diagonal) polynomial F of degree d in d variables. The distribution of such a set S in N is largely conjectural, but in some cases we can prove that there are arbitrarily large intervals contained in the complement, thanks to a special phenomenon in low degree. The proof involves some beautiful mathematics, from Hecke L-functions to the theory of Kummer extensions.

Abstract: I will discuss recent joint work with James Parks and Anders Södergren. Looking at the one-level density of low-lying zeros of quadratic Dirichlet L-functions, Katz and Sarnak predicted a sharp transition in the main terms when the support of the Fourier transform of the implied test functions reaches the point 1. By estimating this quantity up to a power-saving error term, we show that such a transition is also present in lower-order terms. In particular this answers a question of Rudnick coming from the function field analogue. We also show that this transition is also present in the Ratios Conjecture's prediction.

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