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Tuesday, September 1, 2020

**Abstract:** Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally. Much of this work is joint with Bill Banks and Terry Tao.