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SUMMARY:Gaussian distribution of squarefree and B-free numbers in short in
 tervals - Alexander Mangerel (Durham University)
DTSTART:20211116T143000Z
DTEND:20211116T153000Z
UID:TALK162205@talks.cam.ac.uk
CONTACT:Rong Zhou
DESCRIPTION:(Joint with O. Gorodetsky and B. Rodgers) It is a classical qu
 est in analytic number theory to understand the fine-scale distribution of
  arithmetic sequences such as the primes. For a given length scale h\, the
  number of elements of a ``nice'' sequence in a uniformly randomly selecte
 d interval (x\,x+h]\, 1≤x≤X\, might be expected to follow the statisti
 cs of a normally distributed random variable (in suitable ranges of 1 ≤ 
 h ≤ X).  Following the work of Montgomery and Soundararajan\, this is kn
 own to be true for the primes\, but only if we assume several deep and lon
 g-standing conjectures such as the Riemann Hypothesis. In fact\, previousl
 y such distributional results had not been proven for any (non-trivial) se
 quence of number-theoretic interest\, unconditionally.\n\nAs a model for t
 he primes\, in this talk I will address such statistical questions for the
  sequence of squarefree numbers\, i.e.\, numbers not divisible by the squa
 re of any prime\, among other related ``sifted'' sequences called B-free n
 umbers. I hope to further motivate and explain our main result that shows\
 , unconditionally\, that short interval counts of squarefree numbers do sa
 tisfy Gaussian statistics\, answering several old questions of R.R. Hall.
LOCATION:MR13
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