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Berkeley-Caltech-Stanford Joint Number Theory Seminar

Monday, March 28, 2022
12:30pm to 1:30pm
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Expansion, divisibility and parity
Harald Helfgott, Mathematisches Institut, University of Göttingen.,

We will discuss a graph that encodes the divisibility properties of integers by primes. We show that this graph is shown to have a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining our result with Matomaki-Radziwill. For instance: for lambda the Liouville function (that is, the completely multiplicative 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 also manage to prove, for example, that lambda(n+1) averages to 0 at 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<=N). 

Joint work with M. Radziwill.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].