Explicit calculations for Sono's multidimensional sieve of \(E_2\)-numbers (Q6590635)

An \(E_2\)-number is being defined as the product of two distinct primes. By using the main theorem of \textit{K. Sono} [J. Math. Soc. Japan 72, No. 1, 81--118 (2020; Zbl 1473.11181)], and finding explicit expressions for some integrals of certain symmetric polynomials, the authors obtain several results concerning the gaps between \(E_2\)-numbers. In particular, they prove that there are infinitely many intervals of length 94 containing quadruples of \(E_2\)-numbers, and reduce this 94 to 32 under Elliot-Halberstam Hypotheses on primes and on \(E_2\)-numbers. Larger gaps are also investigated. The proof also relies on [Ann. Math. (2) 181, No. 1, 383--413 (2015; Zbl 1306.11073)] by \textit{J. Maynard}. As noted by the authors, the method of the present paper in fact detects \emph{sifted} \(E_2\)-numbers: given parameters \(\eta\in(0,1/4]\) and \(N\ge10\), a \textit{sifted} \(E_2\)-number in \((N,2N]\) is defined as a product \(p_1p_2\in(N,2N]\) where \(N^\eta < p_1\le N^{1/2}< p_2\).