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.

Abstract: We will consider different Diophantine exponents for simultaneous Diophantine approximation and for one linear form and discuss some relations between them. Some of the inequalities are optimal. This follows from D. Roy's theorem about Schmidt-Summerer's graphs. Some of the results are obtained jointly with A. Marnat.

Title: The sixth moment of the Riemann zeta function and ternary additive divisor sums

Abstract: Hardy and Littlewood initiated the study of the 2k-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. In this talk we consider the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula for the sixth moment. This builds on earlier work of Ivic and of Conrey-Gonek.

Abstract: In this talk, we shall present recent results concerning the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as "Birch sums". The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry, and can also be generalized to other types of $\ell$-adic trace functions. In particular, some of our results also hold for partial sums of Kloosterman sums. As an application, we exhibit large values of partial sums of Birch sums, which we believe are best possible.