Analytic Number Theory
Org:
Kevin Hare,
Wentang Kuo and
YuRu Liu (University of Waterloo)
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 KARL DILCHER, Dalhousie University
 DANIEL FIORILLI, University of Ottawa
Lowlying zeros of quadratic Dirichlet $L$functions: the transition [PDF]

I will discuss recent joint work with James Parks and Anders SÃ¶dergren. Looking at the onelevel density of lowlying 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 powersaving error term, we show that such a transition is also present in lowerorder 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
$$\int_{[0,1)^k}\Biggl \sum_{1\le n\le X}e(\alpha_1x+\ldots +\alpha_kx^k)\Biggr^{2s}\,{\rm d}{\underline \alpha}\ll X^{s+\epsilon}+X^{2sk(k+1)/2}.$$
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
© Canadian Mathematical Society