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SUMMARY:Harald Andrés Helfgott (University of Göttingen/Institut de Math
 ématiques de Jussieu)
DTSTART:20230403T163000Z
DTEND:20230403T173000Z
DTSTAMP:20260423T021132Z
UID:NTC/22
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTC/22/">Exp
 ansion\, divisibility and parity</a>\nby Harald Andrés Helfgott (Universi
 ty of Göttingen/Institut de Mathématiques de Jussieu) as part of Lethbri
 dge number theory and combinatorics seminar\n\nLecture held in University 
 of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nWe will discuss a graph t
 hat encodes the divisibility properties of integers by primes. We prove th
 at this graph has a strong local expander property almost everywhere. We t
 hen obtain several consequences in number theory\, beyond the traditional 
 parity barrier\, by combining our result with Matomaki-Radziwill. For inst
 ance: for lambda the Liouville function (that is\, the completely multipli
 cative function with $\\lambda(p) = -1$ for every prime)\, $(1/\\log x) \\
 sum_{n\\leq x} \\lambda(n) \\lambda(n+1)/n = O(1/\\sqrt(\\log \\log x))$\,
  which is stronger than well-known results by Tao and Tao-Teravainen. We a
 lso manage to prove\, for example\, that $\\lambda(n+1)$ averages to $0$ a
 t almost all scales when $n$ restricted to have a specific number of prime
  divisors $\\Omega(n)=k$\, for any "popular" value of $k$ (that is\, $k = 
 \\log \\log N + O(\\sqrt(\\log \\log N)$) for $n \\leq N$).\n
LOCATION:https://researchseminars.org/talk/NTC/22/
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