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SUMMARY:Gál type GCD sums and extreme values of the Riemann zeta function
  - Kristian Seip (NTNU)
DTSTART:20160301T141500Z
DTEND:20160301T151500Z
UID:TALK64260@talks.cam.ac.uk
CONTACT:Jack Thorne
DESCRIPTION:In recent joint work with Aistleitner\, Berkes\, Bondarenko\, 
 and Hilberdink\, we have found optimal bounds for sums over _k_ and _l_ fr
 om _1_ to _N_ of _gcd(nk\,nl)^2S ^ / (nk nl)^S_ where _n1\,...\, nN_ are d
 istinct positive integers and _0 < S < 1_ . Such sums are named after Gál
  who in 1949\, solving a prize problem proposed by Erdos\, settled the cas
 e _S=1_. \nI will discuss the relation between such estimates and extreme 
 values of |zeta(S+it)|. \nIn particular\, I will present the following the
 orem 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 \n_exp(c sqrt{log T logloglog T/ loglog
  T})_ for all T large enough. Our proof relies on Soundararajan's resonanc
 e method. I will outline the main novelties of our adaption of this method
 \, including our usage of large Gál type sums.\n
LOCATION:MR13\, Centre for Mathematical Sciences
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