Research PapersNo Access

Another note on smooth numbers in short intervals

Kaisa Matomäki

Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland

https://doi.org/10.1142/S1793042116500196Cited by:1 (Source: Crossref)

Abstract

In this paper, we prove that, for any positive constants $δ$ and $ε$ and every large enough $x$ , the interval $[x, x + \sqrt{x} {(log x)}^{7 / 3 + δ}]$ contains numbers whose all prime factors are smaller than $x^{ε}$ .

Keywords:

AMSC: 11N25

References

1. A. Balog, On the distribution of integers having no large prime factor, Astérisque 147–148 (1987) 27–31. Google Scholar
2. E. Croot, Smooth numbers in short intervals, Int. J. Number Theory 3 (2007) 159–169. Link, Web of Science, Google Scholar
3. J. B. Friedlander and A. Granville, Smoothing “smooth” numbers, Philos. Trans. Roy. Soc. London Ser. A 345(1676) (1993) 339–347. Crossref, Web of Science, Google Scholar
4. A. Granville, Smooth numbers: Computational number theory and beyond, in Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, MSRI Publications, Vol. 44 (Cambridge University Press, Cambridge, 2008), pp. 267–323. Google Scholar
5. G. Harman, Almost-primes in short intervals, Math. Ann. 258(1) (1981/82) 107–112. Crossref, Web of Science, Google Scholar
6. G. Harman, Short intervals containing numbers without large prime factors, Math. Proc. Cambridge Philos. Soc. 109(1) (1991) 1–5. Crossref, Web of Science, Google Scholar
7. G. Harman, Integers without large prime factors in short intervals and arithmetic progressions, Acta Arith. 91 (1999) 279–289. Crossref, Web of Science, Google Scholar
8. G. Harman, Prime-Detecting Sieves, London Mathematical Society Monographs (New Series), Vol. 33 (Princeton University Press, Princeton, 2007). Google Scholar
9. A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. Théor. Nombres Bordeaux 5(2) (1993) 411–484. Crossref, Google Scholar
10. A. Ivić, The Riemann Zeta-Function. Theory and Applications (Dover Publications, New York, 2003); Reprint of the 1985 original. Google Scholar
11. H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, Vol. 53 (American Mathematical Society, Providence, RI, 2004). Crossref, Google Scholar
12. H. W. Lenstra, Jr.,, J. Pila and C. Pomerance, A hyperelliptic smoothness test. I, Philos. Trans. Roy. Soc. London Ser. A 345(1676) (1993) 397–408. Crossref, Web of Science, Google Scholar
13. K. Matomäki, A note on smooth numbers in short intervals, Int. J. Number Theory 6 (2010) 1113–1116. Link, Web of Science, Google Scholar
14. K. Soundararajan, Smooth numbers in short intervals, preprint (2010); http://arxiv.org as arXiv:1009.1591v1 [math.NT]. Google Scholar
15. D. Wolke, Fast-Primzahlen in kurzen Intervallen, Math. Ann. 244(3) (1979) 233–242. Crossref, Web of Science, Google Scholar