Almost primes in almost all short intervals II

Matomäki, Kaisa; Teräväinen, Joni

doi:10.1090/tran/8869

Almost primes in almost all short intervals II
HTML articles powered by AMS MathViewer

by Kaisa Matomäki and Joni Teräväinen

Trans. Amer. Math. Soc. 376 (2023), 5433-5459

DOI: https://doi.org/10.1090/tran/8869

Published electronically: May 9, 2023

HTML | PDF | Request permission

Abstract:

We show that, for almost all $x$, the interval $(x, x+(\log x)^{2.1}]$ contains products of exactly two primes. This improves on a work of the second author that had $3.51$ in place of $2.1$. To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown’s mean value theorem for sparse Dirichlet polynomials.

References

R. C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes. II, Proc. London Math. Soc. (3) 83 (2001), no. 3, 532–562. MR 1851081, DOI 10.1112/plms/83.3.532
Jean Bourgain, On large values estimates for Dirichlet polynomials and the density hypothesis for the Riemann zeta function, Internat. Math. Res. Notices 3 (2000), 133–146. MR 1741611, DOI 10.1155/S107379280000009X
J.-M. Deshouillers and H. Iwaniec, Power mean values of the Riemann zeta function, Mathematika 29 (1982), no. 2, 202–212 (1983). MR 696876, DOI 10.1112/S0025579300012298
Glyn Harman, Almost-primes in short intervals, Math. Ann. 258 (1981/82), no. 1, 107–112. MR 641671, DOI 10.1007/BF01450349
Glyn Harman, Prime-detecting sieves, London Mathematical Society Monographs Series, vol. 33, Princeton University Press, Princeton, NJ, 2007. MR 2331072
D. R. Heath-Brown, Almost-primes in arithmetic progressions and short intervals, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 3, 357–375. MR 491558, DOI 10.1017/S0305004100054657
D. R. Heath-Brown, The number of primes in a short interval, J. Reine Angew. Math. 389 (1988), 22–63. MR 953665, DOI 10.1515/crll.1988.389.22
D. R. Heath-Brown, The differences between consecutive smooth numbers, Acta Arith. 184 (2018), no. 3, 267–285. MR 3842905, DOI 10.4064/aa170913-11-7
Adolf Hildebrand and Gérald Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5 (1993), no. 2, 411–484. MR 1265913, DOI 10.5802/jtnb.101
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
Chaohua Jia, Almost all short intervals containing prime numbers, Acta Arith. 76 (1996), no. 1, 21–84. MR 1390568, DOI 10.4064/aa-76-1-21-84
M. Jutila, On large values of Dirichlet polynomials, Topics in number theory (Proc. Colloq., Debrecen, 1974) Colloq. Math. Soc. János Bolyai, Vol. 13, North-Holland, Amsterdam-Oxford-New York, 1976, pp. 129–140. MR 434989
Kaisa Matomäki, Almost primes in almost all very short intervals, J. Lond. Math. Soc. (2) 106 (2022), no. 2, 1061–1097. MR 4477211, DOI 10.1112/jlms.12592
Kaisa Matomäki and Maksym Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. (2) 183 (2016), no. 3, 1015–1056. MR 3488742, DOI 10.4007/annals.2016.183.3.6
Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
Yoichi Motohashi, A note on almost-primes in short intervals, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 6, 225–226. MR 542398
Atle Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), no. 6, 87–105. MR 12624
Joni Teräväinen, Almost primes in almost all short intervals, Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 2, 247–281. MR 3530507, DOI 10.1017/S0305004116000232
N. Watt, Kloosterman sums and a mean value for Dirichlet polynomials, J. Number Theory 53 (1995), no. 1, 179–210. MR 1344840, DOI 10.1006/jnth.1995.1086
Dieter Wolke, Fast-Primzahlen in kurzen Intervallen, Math. Ann. 244 (1979), no. 3, 233–242 (German). MR 553254, DOI 10.1007/BF01420345