Explicit Calculations for Sono's Multidimensional Sieve of $E_2$-Numbers

TL;DR

This paper derives explicit formulas for integrals related to symmetric polynomials in Sono's multidimensional sieve of $E_2$-numbers, leading to improved bounds on gaps between such numbers, with some results conditional on the Elliott-Halberstam conjecture.

Contribution

It provides explicit integral formulas for symmetric polynomials in the sieve and establishes new bounds on gaps between $E_2$-numbers, including unconditional and conditional results.

Findings

Infinitely many $E_2$-numbers within a gap of 94 unconditionally.

Within a gap of 32 assuming Elliott-Halberstam conjecture.

Improved bounds on the distribution of $E_2$-numbers.

Abstract

We derive explicit formulas for integrals of certain symmetric polynomials used in Keiju Sono's multidimensional sieve of $E_2$-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 $E_2$-numbers. For example, we show there are infinitely many occurrences of four $E_2$-numbers within a gap size of 94 unconditionally and within a gap size of 32 assuming the Elliott-Halberstam conjecture for primes and sifted $E_2$-numbers.