2017 CMS Winter Meeting

Waterloo, December 8 - 11, 2017

Analytic Number Theory
Org: Kevin Hare, Wentang Kuo and Yu-Ru Liu (University of Waterloo)
[PDF]

KARL DILCHER, Dalhousie University

DANIEL FIORILLI, University of Ottawa
Low-lying zeros of quadratic Dirichlet $L$-functions: the transition  [PDF]

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.

TRISTAN FREIBERG, University of Waterloo

JOHN FRIEDLANDER, University of Toronto

ALIA HAMIEH, University of Northern British Columbia

MATILDE LALIN, University of Montreal

YOUNESS LAMZOURI, York University

ALLYSA LUMLEY, York University

GREG MARTIN, University of British Columbia

RAM MURTY, Queen's University

JONATHAN SORENSON, Butler University

AKSHAA VATWANI, University of Waterloo

TREVOR WOOLEY, University of Bristol
Nested efficient congruencing and relatives of Vinogradov's mean value theorem  [PDF]

The main conjecture in Vinogradov's mean value theorem states that, for each $\epsilon>0$, one has $$\int_{[0,1)^k}\Biggl| \sum_{1\le n\le X}e(\alpha_1x+\ldots +\alpha_kx^k)\Biggr|^{2s}\,{\rm d}{\underline \alpha}\ll X^{s+\epsilon}+X^{2s-k(k+1)/2}.$$ This is now a theorem of Bourgain, Demeter and Guth (in 2016, via $l^2$-decoupling) and the speaker (in 2014 for k=3, and in 2017 in general, via (nested) efficient congruencing). We report on some generalisations of this conclusion, some of which go beyond the orbit of decoupling and efficient congruencing.

XIAOMEI ZHAO, Central China Normal University and Waterloo