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SUMMARY:Pieter Moree (MPI Bonn)
DTSTART:20210923T130000Z
DTEND:20210923T140000Z
DTSTAMP:20260423T022841Z
UID:UCDANT/15
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCDANT/15/">
 Euler-Kronecker constants and cusp forms</a>\nby Pieter Moree (MPI Bonn) a
 s part of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nRamanuja
 n\, in a manuscript not published during his lifetime\, made a very precis
 e conjecture for how many integers $n\\le x$ the Ramanujan tau function $\
 \tau(n)$ is not divisible by 691 (and likewise for some other primes). The
  $\\tau(n)$ are the Fourier coefficients of the Delta function\, which is 
 a cusp form for the full modular group. Rankin proved that Ramanujan's cla
 im is asymptotically correct. However\, the speaker showed in 2004 that th
 e second-order behavior predicted by Ramanujan does not match reality. The
  proof makes use of high precision computation of constants akin to the Eu
 ler-Mascheroni constant called Euler-Kronecker constants.\n\nRecently the 
 author\, joint with Ciolan and Languasco\, studied the analogue of Ramanuj
 an's conjecture for the exceptional primes\, as classified by Serre and Sw
 innerton-Dyer\, for the 5 cups forms akin to the Delta function for which 
 the space of cusp forms is 1-dimensional. The tool for this is a high-prec
 ision evaluation of the number of integers  $n\\le x$ for which a prescrib
 ed integer $q$ does not divide the $k$th sum of divisors function\, sharpe
 ning earlier work of Rankin and Scourfield. In my talk\, I will report on 
 generalities on Euler-Kronecker constants and the above work\, with ample 
 of historical material.\n
LOCATION:https://researchseminars.org/talk/UCDANT/15/
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