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Philadelphia Area Number Theory Seminar

Wednesday, October 24, 2018 - 2:15pm

Raphael Steiner

Institute for Advanced Study

Location

Bryn Mawr College

Park Science Building, Room 328

Talk is scheduled to begin at 2:40 PM. Tea and refreshments will be served at 2:20 PM in the Math Lounge, Park Science Building, Room 361.

It is a classical theorem in the theory of modular forms that the points x/ N, where x ∈ Z runs over all the solutions to i=1 xi = N, equidistribute on Sn−1 for n  4 as N (odd) tends to infinity. The rate of equidistribution poses however a more challenging problem. Due to its Diophantine nature the points inherit a repulsion property, which opposes equidistribution on small sets. Sarnak conjectures that this Diophantine repulsion is the only obstruction to the rate of equidistribution. Using the smooth delta-symbol circle method, developed by Heath-Brown, Sardari was able to show that the conjecture is true for n  5 and recovering Sarnak’s progress towards the conjecture for n = 4. Building on Sardari’s work, Browning, Kumaraswamy, and myself were able to reduce the conjecture to correlation sums of Kloosterman sums of the following type: q≤Q 1/q S(m, n; q) exp(4πiα√mn/q). Assuming the twisted Linnik conjecture, which states that the above sum is O((Qmn)) for |α|  2, we are able to verify Sarnak’s Conjecture. I shall lose a few words on the unconditional progress towards this conjecture and how (unfortunately) it is insufficient to improve unconditionally what is known towards Sarnak’s conjecture. If time permits, I will talk about ongoing research of how the automorphic approach and the circle method approach may be combined to hopefully give better insight into Sarnak’s conjecture.