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SUMMARY:Divisor distribution of random integers. -  Ben Green (Oxford)
DTSTART:20220519T133000Z
DTEND:20220519T143000Z
UID:TALK173945@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION: Let n be a random integer (sampled from {1\,..\,X} for some l
 arge\nX). It is a classical fact that\, typically\, n will have around (lo
 g n)^{log\n2} divisors. Must some of these be close together? Hooley's Del
 ta function\nDelta(n) is the maximum\, over all dyadic intervals I = [t\,2
 t]\, of the\nnumber of divisors of n in I. I will report on joint work wit
 h Kevin Ford\nand Dimitris Koukoulopoulos where we conjecture that typical
 ly Delta(n) is\nabout (log log n)^c for some c = 0.353.... given by an equ
 ation involving\nan exotic recurrence relation\, and then prove (in some s
 ense) half of this\nconjecture\, establishing that Delta(n) is at least th
 is big almost surely.\nFor the most part I will discuss a model combinator
 ial problem about\nrepresenting integers in many ways as sums of elements 
 from a random set.
LOCATION:MR12
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