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I've been learning various basics of Sieve Methods in Analytic Number Theory, and I'm wondering what are some uses of these methods in current research? Not famous, unsolved problems, but areas of research currently underway? I understand sieve methods' uses in the context of enumerating primes, even pairs of primes (Ex. recasting the twin primes conjecture as a sieve problem). But what other, current, applications could there be?

Thanks!

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A recent application was the reduction of the prime gap. The idea was to show that, $$ \liminf_n p_{n+1}-p_n < \infty $$ where $p_n$ denotes the $n^{th}$ prime number. Yitang Zhang managed to prove that indeed the prime gap is bounded by some finite number ($70,000,000$) in 2013 in a remarkable breakthrough. Subsequently, Terence Tao and James Maynard among others improved this bound to $246$ using what is known as the Selberg Sieve. The argument is pretty novel in Sieve Methods and is a definite read for anyone interested in learning about sieves.

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