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Gál type GCD sums and extreme values of the Riemann zeta function

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In recent joint work with Aistleitner, Berkes, Bondarenko, and Hilberdink, we have found optimal bounds for sums over k and l from 1 to N of gcd(nk,nl)2S / (nk nl)^S where n1,..., nN are distinct positive integers and 0 < S < 1 . Such sums are named after Gál who in 1949, solving a prize problem proposed by Erdos, settled the case S=1. I will discuss the relation between such estimates and extreme values of |zeta(S+it)|. In particular, I will present the following theorem of Bondarenko and myself: For every c with 0 < c < 1/sqrt{2} there exists a B with 0 < B < 1 such that the maximum of |zeta(1/2+it)| for t between T to the B and T exceeds exp(c sqrt{log T logloglog T/ loglog T}) for all T large enough. Our proof relies on Soundararajan’s resonance method. I will outline the main novelties of our adaption of this method, including our usage of large Gál type sums.

This talk is part of the Number Theory Seminar series.

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