Skip to main content Accessibility help
×
Home

Large prime factors on short intervals

  • JORI MERIKOSKI (a1)

Abstract

We show that for all large enough x the interval [x, x + x1/2 log1.39x] contains numbers with a prime factor p > x18/19. Our work builds on the previous works of Heath–Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals [x, x + x1/2 + ϵ]. We also incorporate some ideas from Harman’s book Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomäki and Radziwiłł (2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of log x when applying Harman’s sieve method.

Copyright

References

Hide All
[1] Baker, R. and Harman, G.. Numbers with a large prime factor. II. In Analytic number theory (Cambridge Univ. Press, Cambridge, 2009), pages 114.
[2] Baker, R. C., Harman, G. and Pintz, J.. The difference between consecutive primes. II. Proc. London Math. Soc. (3), 83(3) (2001), 532562.
[3] Balog, A.. Numbers with a large prime factor. Studia Sci. Math. Hungar. 15(1-3) (1980), 139146.
[4] Balog, A.. Numbers with a large prime factor. II. In Topics in classical number theory, Vol. I, II (Budapest, 1981), volume 34 of Colloq. Math. Soc. János Bolyai, (North-Holland, Amsterdam, 1984), pages 4967.
[5] Balog, A., Harman, G. and Pintz, J.. Numbers with a large prime factor III. Quart. J. Math. Oxford Ser. (2), 34(134) (1983), 133140.
[6] Balog, A., Harman, G. and Pintz, J.. Numbers with a large prime factor. IV. J. London Math. Soc. (2), 28(2) (1983), 218226.
[7] Friedlander, J. and Iwaniec, H.. Opera de cribro, volume 57 of American Mathematical Society Colloquium Publication (American Mathematical Society, Providence, RI, 2010).
[8] Harman, G.. Prime-detecting sieves, volume 33 of London Mathematical Society Monographs Serie. (Princeton University Press, Princeton, NJ, 2007).
[9] Heath-Brown, D. R.. The largest prime factor of the integers in an interval. Sci. China Ser. A, 39(5) (1996), 449476.
[10] Heath-Brown, D. R. and Jia, C.. The largest prime factor of the integers in an interval II. J. Reine Angew. Math. 498 (1998), 3559.
[11] Iwaniec, H. and Kowalski, E.. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications (American Mathematical Society, Providence, RI, 2004).
[12] Jia, C. and Liu, M.C.. On the largest prime factor of integers. Acta Arith . 95(1) (2000), 1748.
[13] Jutila, M.. On numbers with a large prime factor. J. Indian Math. Soc. (N.S.) 37 (1973/74), 4353.
[14] Matomäki, K.. Another note on smooth numbers in short intervals. Int. J. Number Theory 12(2) (2016), 323340.
[15] Matomäki, K. and Radziwiłł, M.. Multiplicative functions in short intervals. Ann. of Math. (2), 183(3) (2016), 10151056.
[16] Shiu, P.. A Brun-Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313 (1980), 161170.
[17] Tao, T.. 254a, supplement 3: the gamma function and the functional equation (optional) (2014). URL (Accessed 15.2.2018): https://terrytao.wordpress.com/2014/12/15/254a-supplement-3-the-gamma-function-and-the-functional-equation-optional/.
[18] Tenenbaum, G.. Introduction to analytic and probabilistic number theory, volume 163 of Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, third edition, 2015). Translated from the 2008 French edition by Patrick D. F. Ion.
[19] Teräväinen, J.. Almost primes in almost all short intervals. Math. Proc. Camb. Phil. Soc. 161(2) (2016), 247281.

MSC classification

Large prime factors on short intervals

  • JORI MERIKOSKI (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed