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SUMMARY:Ordering random variables and racing prime numbers - Adam Harper (
 Cambridge)
DTSTART:20160216T163000Z
DTEND:20160216T173000Z
UID:TALK64519@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:The ``prime number race'' is the competition between different
  coprime residue classes mod $q$ to contain the most primes\, up to a poin
 t $x$. Rubinstein and Sarnak showed\, assuming two number theory conjectur
 es\, that as $x$ varies the problem is equivalent to a problem about order
 ings of approximately Gaussian random variables\, having weak correlations
  coming from number theory. In particular\, as $q \\rightarrow \\infty$ th
 e number of primes in any fixed set of $r$ coprime classes will achieve an
 y given ordering for $\\sim 1/r!$ values of $x$. I will try to explain wha
 t happens when $r$ is allowed to grow as a function of $q$. It turns out t
 hat one still sees uniformity of orderings in many situations\, but not al
 ways. The proofs involve various probabilistic ideas\, and also some harmo
 nic analysis related to the Hardy--Littlewood method. This is joint work w
 ith Youness Lamzouri.\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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