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

Goldston, Daniel; Panidapu, Apoorva; Schettler, Jordan

doi:10.1090/mcom/3938

Explicit calculations for Sono’s multidimensional sieve of $E_2$-numbers
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by Daniel A. Goldston, Apoorva Panidapu and Jordan Schettler;

Math. Comp. 93 (2024), 2943-2958

DOI: https://doi.org/10.1090/mcom/3938

Published electronically: January 23, 2024

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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 $E_2$-numbers.

References

Enrico Bombieri, Le grand crible dans la théorie analytique des nombres, Astérisque 18 (1987), 103 (French, with English summary). MR 891718
P. D. T. A. Elliott and H. Halberstam, A conjecture in prime number theory, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London-New York, 1970, pp. 59–72. MR 276195
D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım, Small gaps between products of two primes, Proc. Lond. Math. Soc. (3) 98 (2009), no. 3, 741–774. MR 2500871, DOI 10.1112/plms/pdn046
Daniel A. Goldston, Apoorva Panidapu, and Jordan Schettler, Small gaps between three almost primes and almost prime powers, Acta Arith. 203 (2022), no. 1, 1–18. MR 4415993, DOI 10.4064/aa210325-27-11
Daniel A. Goldston, János Pintz, and Cem Y. Yıldırım, Primes in tuples. I, Ann. of Math. (2) 170 (2009), no. 2, 819–862. MR 2552109, DOI 10.4007/annals.2009.170.819
James Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), no. 1, 383–413. MR 3272929, DOI 10.4007/annals.2015.181.1.7
Yoichi Motohashi, An induction principle for the generalization of Bombieri’s prime number theorem, Proc. Japan Acad. 52 (1976), no. 6, 273–275. MR 422179
D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Res. Math. Sci. 1 (2014), Art. 12, 83. MR 3373710, DOI 10.1186/s40687-014-0012-7
Keiju Sono, Small gaps between the set of products of at most two primes, J. Math. Soc. Japan 72 (2020), no. 1, 81–118. MR 4055091, DOI 10.2969/jmsj/74937493
A. I. Vinogradov, The density hypothesis for Dirichet $L$-series, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903–934 (Russian). MR 197414
Yitang Zhang, Bounded gaps between primes, Ann. of Math. (2) 179 (2014), no. 3, 1121–1174. MR 3171761, DOI 10.4007/annals.2014.179.3.7