Explicit calculations for Sono’s multidimensional sieve of 𝐸₂-numbers

Abstract

We derive explicit formulas for integrals of certain symmetric polynomials used in Keiju Sono’s multidimensional sieve of E 2 E_2 -numbers, i.e., integers which are products of two distinct primes. We use these computations to produce the currently best-known bounds for gaps between multiple E 2 E_2 -numbers. For example, we show there are infinitely many occurrences of four E 2 E_2 -numbers within a gap size of 94 94 unconditionally and within a gap size of 32 32 assuming the Elliott-Halberstam conjecture for primes and E 2 E_2 -numbers.