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SUMMARY:Expansion\, divisibility and parity - Harald Helfgott (CNRS--Jussi
 eu)
DTSTART:20260325T133000Z
DTEND:20260325T143000Z
UID:TALK242041@talks.cam.ac.uk
CONTACT:Julia Wolf
DESCRIPTION:We will discuss a graph that encodes the divisibility properti
 es of integers by primes. We will prove that this graph has a strong local
  expander property almost everywhere. We then obtain several consequences 
 in number theory\, beyond the traditional parity barrier\, by combining th
 e main result with Matomaki-Radziwill. (This is joint work with M. Radziwi
 ll.) \n\nFor instance: for lambda the Liouville function (that is\, the co
 mpletely multiplicative function with lambda(p) = -1 for every prime)\, (1
 /\\log x) sum_{n≤x} lambda(n) lambda(n+1)/n = O(1/sqrt(log log x))\, whi
 ch is stronger than well-known results by Tao and Tao-Teravainen. \n\nWe a
 lso manage to prove\, for example\, that lambda(n+1) averages to 0 at almo
 st all scales when n restricted to have a specific number of prime divisor
 s Omega(n)=k\, for any "popular" value of k (that is\, k = log log N + O(s
 qrt(log log N)) for n≤N).\n\nWe shall also discuss a recent generalizati
 on by C. Pilatte\, who has succeeded in proving that a graph with edges th
 at are rough integers\, rather than primes\, also has a strong local expan
 der property almost everywhere\, following the same strategy. As a result\
 , he has obtained a bound with O(1/(log x)^c) instead of O(1/sqrt(log log 
 x)) in the above\, as well as other improvements in consequences across th
 e board.
LOCATION:MR5\, CMS
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