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Sieve weights and their smoothings

Connections for Women: Analytic Number Theory February 02, 2017 - February 03, 2017

February 03, 2017 (09:15 AM PST - 10:15 AM PST)
Speaker(s): Dimitris Koukoulopoulos (Université de Montréal)
Location: MSRI: Simons Auditorium
Tags/Keywords
  • Mobius function moments

  • divisor sums

  • sieve methods

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Sieve Weights And Their Smoothings

Abstract

I will discuss moments of partially smoothed, truncated divisor sums of the M\"obius function. Such divisor sums appear naturally in the theory of the Selberg sieve and they play a key role in the GPY sieve and its recent improvements due to Maynard and Tao. It turns out that if the truncation is smooth enough, the main contribution to the moments comes from almost primes. However, for rougher truncations, the dominant contribution comes from integers with many prime factors. Analogous questions can be asked for polynomials over finite fields and for permutations, and in these cases the moments behave rather differently, a rare exception. As we will see, a plausible explanation for this phenomenon is given by studying the analogous sums for Dirichlet characters and obtaining different answers depending on whether or not the character is ``exceptional''. This is joint work with Andrew Granville and James Maynard

Supplements
27892?type=thumb Koukoulopoulous Notes 180 KB application/pdf Download
Video/Audio Files

Sieve Weights And Their Smoothings

H.264 Video 06-Koukoulopoulos.mp4 643 MB video/mp4 rtsp://videos.msri.org/data/000/027/804/original/06-Koukoulopoulos.mp4 Download
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