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SUMMARY:Harald Helfgott (Universitat Gottigen)
DTSTART:20200605T170000Z
DTEND:20200605T172500Z
DTSTAMP:20260423T011224Z
UID:CANT2020/63
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2020/63/
 ">Optimality of the logarithmic upper-bound sieve\, with explicit estimate
 s</a>\nby Harald Helfgott (Universitat Gottigen) as part of Combinatorial 
 and additive number theory (CANT 2021)\n\n\nAbstract\nAt the simplest leve
 l\, an upper bound sieve of Selberg type is a choice of $\\rho(d)$\, $d\\l
 eq D$\, with $\\rho(1)=1$\, such that\n$$S = \\sum_{n\\leq N} \\left(\\sum
 _{d|n} \\mu(d) \\rho(d)\\right)^2$$\nis as small as possible.\nThe optimal
  choice of $\\rho(d)$ for given $D$ was found by Selberg. However\, for se
 veral applications\, it is better to work with functions $\\rho(d)$ that a
 re scalings of a given continuous or monotonic function $\\eta$. The quest
 ion is then: What is the best function $\\eta$\, and how does $S$ for give
 n $\\eta$ and $D$ compare to $S$ for Selberg's choice? \n\nThe most common
  choice of $\\eta$ is that of Barban-Vehov (1968)\, which gives an $S$ wit
 h the same main term as Selberg's $S$. We show that Barban and Vehov's cho
 ice 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 term\, which is
  negative and which we determine explicitly. \n\nJoint work with Emanuel C
 arneiro\, Andrés Chirre and Julian Mejía-Cordero.\n
LOCATION:https://researchseminars.org/talk/CANT2020/63/
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