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

Daniel A. Goldston; Apoorva Panidapu; Jordan Schettler

paper

Explicit Calculations for Sono's Multidimensional Sieve of $E_{2}$ -Numbers

arXiv:2208.13931

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.

15 pages

Explicit Calculations for Sono's Multidimensional Sieve of E2​-Numbers

Abstract

Explicit Calculations for Sono's Multidimensional Sieve of $E_{2}$ -Numbers