[Liste-CICMA] Un-QVNTS, Maksym Radziwill (Thursday, February 5)

Guillermo Martinez-Zalce martinez at CRM.UMontreal.CA
Mon Feb 2 09:28:30 EST 2015


Title: Around the Moebius function

Abstract: The Moebius function plays a central role in number theory; 
both the prime number theorem and the Riemann Hypothesis are naturally 
formulated in terms of the amount of cancellations one gets when summing 
the Moebius function. In recent joint work with K. Matomaki we have 
shown that the sum of the Moebius function exhibits cancellations in 
``almost all intervals'' of increasing length. This goes beyond what was 
previously known even conditionally on the Riemann Hypothesis and allows 
us to settle a long-standing conjecture on correlations of consecutive 
values of the Moebius function. Our result holds in fact in greater 
generality. Exploiting this generality we show that between a fixed 
number of consecutive squares there is always an integer composed of 
only ``small'' prime factors. This settles and old problem on ``smooth 
numbers'' and is related to the running time of Lenstra's factoring 
algorithm. Finally, in recent on-going work with K. Matomaki and T. Tao 
we have been able to use the previously-mentioned general result to show 
that Chowla's conjecture (on correlations of the Moebius function) holds 
on average and we strengthened previous results on patterns in the 
Liouville function (a close cousin of the Moebius function). The talk 
will be targeted towards non-specialists. I will explain in detail the 
meaning, motivation and origins of the above problems and conjectures.

When: February 5, 2014, 10:30-11:30

Where: McGill University, Burnside Hall, Room BH920


More info:

http://www.dms.umontreal.ca/~qvnts/index.html
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