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CATEGORIES:Number Theory Seminar
SUMMARY:Gaussian distribution of squarefree and B-free num
bers in short intervals - Alexander Mangerel (Durh
am University)
DTSTART;TZID=Europe/London:20211116T143000
DTEND;TZID=Europe/London:20211116T153000
UID:TALK162205AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/162205
DESCRIPTION:(Joint with O. Gorodetsky and B. Rodgers) It is a
classical quest in analytic number theory to under
stand the fine-scale distribution of arithmetic se
quences such as the primes. For a given length sca
le h\, the number of elements of a ``nice'' sequen
ce in a uniformly randomly selected interval (x\,x
+h]\, 1≤x≤X\, might be expected to follow the stat
istics of a normally distributed random variable (
in suitable ranges of 1 ≤ h ≤ X). Following the w
ork of Montgomery and Soundararajan\, this is know
n to be true for the primes\, but only if we assum
e several deep and long-standing conjectures such
as the Riemann Hypothesis. In fact\, previously su
ch distributional results had not been proven for
any (non-trivial) sequence of number-theoretic int
erest\, unconditionally.\n\nAs a model for the pri
mes\, in this talk I will address such statistical
questions for the sequence of squarefree numbers\
, i.e.\, numbers not divisible by the square of an
y prime\, among other related ``sifted'' sequences
called B-free numbers. I hope to further motivate
and explain our main result that shows\, uncondit
ionally\, that short interval counts of squarefree
numbers do satisfy Gaussian statistics\, answerin
g several old questions of R.R. Hall.
LOCATION:MR13
CONTACT:Rong Zhou
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