We show that, for almost all x x , the interval ( x , x + ( log x ) 2.1 ] (x, x+(\log x)^{2.1}] contains products of exactly two primes. This improves on a work of the second author that had 3.51 3.51 in place of 2.1 2.1 . To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown’s mean value theorem for sparse Dirichlet polynomials.
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Almost exact exchange at almost no computational cost in electronic structure
