Seminar series
Date
Thu, 05 May 2022
Time
16:00 -
17:00
Location
L5
Speaker
Alexander Mangerel
Organisation
Durham University
(Joint with O. Gorodetsky and B. Rodgers) It is a classical quest 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 selected interval $(x,x+h], 1 \leq x \leq X$, might be expected to follow the statistics of a normally distributed random variable (in suitable ranges of $1 \leq h \leq X$). Following the work of Montgomery and Soundararajan, this is known to be true for the primes, but only if we assume several deep and long-standing conjectures among which the Riemann Hypothesis. In fact, previously such distributional results had not been proven for any (non-trivial) sequence of number-theoretic interest, unconditionally.
As a model for the primes, in this talk I will address such statistical questions for the sequence of squarefree numbers, i.e., numbers not divisible by the square of any prime, among other related ``sifted'' sequences called B-free numbers. I hope to further motivate and explain our main result that shows, unconditionally, that short interval counts of squarefree numbers do satisfy Gaussian statistics, answering several questions of R.R. Hall.