Divisor distribution of random integers.

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• Ben Green (Oxford)
• Thursday 19 May 2022, 14:30-15:30
• MR12.

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Let n be a random integer (sampled from {1,..,X} for some large X). It is a classical fact that, typically, n will have around (log n) ^{divisors. Must some of these be close together? Hooley’s Delta function Delta(n) is the maximum, over all dyadic intervals I = [t,2t], of the number of divisors of n in I. I will report on joint work with Kevin Ford and Dimitris Koukoulopoulos where we conjecture that typically Delta(n) is about (log log n)}c for some c = 0.353…. given by an equation involving an exotic recurrence relation, and then prove (in some sense) half of this conjecture, establishing that Delta(n) is at least this big almost surely. For the most part I will discuss a model combinatorial problem about representing integers in many ways as sums of elements from a random set.

This talk is part of the Combinatorics Seminar series.

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