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arXiv · 2208.13931

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

Daniel A. Goldston·Apoorva Panidapu·Jordan Schettler
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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.

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Daniel A. Goldston, Apoorva Panidapu, Jordan Schettler. 2022-08-29. Explicit Calculations for Sono's Multidimensional Sieve of $E_2$-Numbers. https://arxiv.org/abs/2208.13931

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