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SUMMARY:Harald Helfgott (Univ. Gottingen/CNRS/Inst. Math. Jussieu)
DTSTART:20200522T110000Z
DTEND:20200522T120000Z
DTSTAMP:20260423T021401Z
UID:NTdL/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/4/">Opt
 imality of the logarithmic upper-bound sieve\, with explicit estimates (jo
 int with Emanuel Carneiro\, Andrés Chirre and Julian Mejía-Cordero)</a>\
 nby Harald Helfgott (Univ. Gottingen/CNRS/Inst. Math. Jussieu) as part of 
 Number theory during lockdown\n\n\nAbstract\nAt the simplest level\, an up
 per bound sieve of Selberg type is a choice of rho(d)\, d<=D\, with rho(1)
 =1\, such that\n\nS = \\sum_{n\\leq N} \\left(\\sum_{d|n} \\mu(d) \\rho(d)
 \\right)^2\n\nis as small as possible.\n\nThe optimal choice of rho(d) for
  given D was found by Selberg. However\, for several applications\, it is 
 better to work with functions rho(d) that are scalings of a given continuo
 us or monotonic function eta. The question is then what is the best functi
 on eta\, and how does S for given eta and D compares to S for Selberg's ch
 oice.\n\nThe most common choice of eta is that of Barban-Vehov (1968)\, wh
 ich gives an S with the same main term as Selberg's S. We show that Barban
  and Vehov's choice is optimal among all eta\, not just (as we knew) when 
 it comes to the main term\, but even when it comes to the second-order ter
 m\, which is negative and which we determine explicitly.\n
LOCATION:https://researchseminars.org/talk/NTdL/4/
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