Joni Teräväinen: Almost Primes in Almost All Short Intervals

When considering \(E_k\) numbers (products of exactly k primes), it is natural to ask, how they are distributed in short intervals. One can show much stronger results when one restricts to almost all intervals. In this context, the question becomes, what is the smallest value of c such that the intervals \([x,x+(\log x)^c] \)contain an \(E_k\) number almost always. Harman showed in 1982 that \(c=7+\varepsilon\) is admissible for \(E_2\) numbers, and this was also the best known result for \(E_k\) numbers with \(k>2\).

We show that for \(E_3\) numbers one can take \(c=1+\varepsilon\), which is optimal up to \(\varepsilon\). We also prove the value \(c=3.51\) for \(E_2\) numbers. The proof uses pointwise, large value and mean value results for Dirichlet polynomials, as well as sieve methods.