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Large prime factors on short intervals

Part of: Multiplicative number theory

Published online by Cambridge University Press:  05 September 2019

JORI MERIKOSKI
JORI MERIKOSKI*
Affiliation:
Department of Mathematics, University of Turku, FI-20014 University of Turku, Finland e-mail: [email protected]
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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.

MSC classification

Primary: 11N05: Distribution of primes
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 1 , January 2021 , pp. 1 - 50
Copyright
© Cambridge Philosophical Society 2019

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References

REFERENCES

Baker, R. and Harman, G.. Numbers with a large prime factor. II. In Analytic number theory (Cambridge Univ. Press, Cambridge, 2009), pages 1–14.Google Scholar
Baker, R. C., Harman, G. and Pintz, J.. The difference between consecutive primes. II. Proc. London Math. Soc. (3), 83(3) (2001), 532–562.CrossRefGoogle Scholar
Balog, A.. Numbers with a large prime factor. Studia Sci. Math. Hungar. 15(1-3) (1980), 139–146.Google Scholar
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 49–67.Google Scholar
Balog, A., Harman, G. and Pintz, J.. Numbers with a large prime factor III. Quart. J. Math. Oxford Ser. (2), 34(134) (1983), 133–140.CrossRefGoogle Scholar
Balog, A., Harman, G. and Pintz, J.. Numbers with a large prime factor. IV. J. London Math. Soc. (2), 28(2) (1983), 218–226.CrossRefGoogle Scholar
Friedlander, J. and Iwaniec, H.. Opera de cribro, volume 57 of American Mathematical Society Colloquium Publication (American Mathematical Society, Providence, RI, 2010).CrossRefGoogle Scholar
Harman, G.. Prime-detecting sieves, volume 33 of London Mathematical Society Monographs Serie. (Princeton University Press, Princeton, NJ, 2007).Google Scholar
Heath-Brown, D. R.. The largest prime factor of the integers in an interval. Sci. China Ser. A, 39(5) (1996), 449–476.Google Scholar
Heath-Brown, D. R. and Jia, C.. The largest prime factor of the integers in an interval II. J. Reine Angew. Math. 498 (1998), 35–59.CrossRefGoogle Scholar
Iwaniec, H. and Kowalski, E.. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
Jia, C. and Liu, M.C.. On the largest prime factor of integers. Acta Arith . 95(1) (2000), 17–48.CrossRefGoogle Scholar
Jutila, M.. On numbers with a large prime factor. J. Indian Math. Soc. (N.S.) 37 (1973/74), 43–53.Google Scholar
Matomäki, K.. Another note on smooth numbers in short intervals. Int. J. Number Theory 12(2) (2016), 323–340.CrossRefGoogle Scholar
Matomäki, K. and Radziwiłł, M.. Multiplicative functions in short intervals. Ann. of Math. (2), 183(3) (2016), 1015–1056.CrossRefGoogle Scholar
Shiu, P.. A Brun-Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313 (1980), 161–170.Google Scholar
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/.Google Scholar
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.CrossRefGoogle Scholar
Teräväinen, J.. Almost primes in almost all short intervals. Math. Proc. Camb. Phil. Soc. 161(2) (2016), 247–281.CrossRefGoogle Scholar