Almost primes in almost all very short intervals

Abstract: By probabilistic models one expects that, as soon as $h \to \infty$ with $X \to \infty$, short intervals of the type $(x- h \log X, x]$ contain primes for almost all $x \in (X/2, X]$. However, this is far from being established. In the talk I discuss related questions and in particular describe how to prove the above claim when one is satisfied with finding $P_2$-numbers (numbers that have at most two prime factors) instead of primes.

algebraic geometrynumber theory

Audience: researchers in the topic

( slides )

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MIT number theory seminar

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