On the largest prime factors of consecutive integers in short intervals
HTML articles powered by AMS MathViewer
- by Zhiwei Wang PDF
- Proc. Amer. Math. Soc. 145 (2017), 3211-3220 Request permission
Abstract:
For an integer $n>1$, let $P(n)$ be the largest prime factor of $n$. We prove that, for $x\rightarrow \infty$, there exists a positive proportion of consecutive integers $n$ and $n+1$ such that $P(n)<P(n+1)$ in short intervals $(x, x+y]$ with $x^{7/12}<y\leqslant x.$ In particular, we have \[ \big |\{n\leqslant x: P(n)< P(n+1)\}\big |> 0.1063 x. \] This improves a previous result of La Bretèche, Pomerance and Tenenbaum.References
- Régis de la Bretèche, Carl Pomerance, and Gérald Tenenbaum, Products of ratios of consecutive integers, Ramanujan J. 9 (2005), no. 1-2, 131–138. MR 2166384, DOI 10.1007/s11139-005-0831-7
- K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr. Fys. 22(A10) (1930), 1–14.
- Paul Erdős and Carl Pomerance, On the largest prime factors of $n$ and $n+1$, Aequationes Math. 17 (1978), no. 2-3, 311–321. MR 480303, DOI 10.1007/BF01818569
- Adolf Hildebrand, On the number of positive integers $\leq x$ and free of prime factors $>y$, J. Number Theory 22 (1986), no. 3, 289–307. MR 831874, DOI 10.1016/0022-314X(86)90013-2
- Henryk Iwaniec, A new form of the error term in the linear sieve, Acta Arith. 37 (1980), 307–320. MR 598883, DOI 10.4064/aa-37-1-307-320
- Henryk Iwaniec, Rosser’s sieve, Acta Arith. 36 (1980), no. 2, 171–202. MR 581917, DOI 10.4064/aa-36-2-171-202
- Jean-Louis Nicolas, Nombres hautement composés, Acta Arith. 49 (1988), no. 4, 395–412 (French). MR 937935, DOI 10.4064/aa-49-4-395-412
- A. Perelli, J. Pintz, and S. Salerno, Bombieri’s theorem in short intervals, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 4, 529–539. MR 808422
- N. M. Timofeev, Distribution of arithmetic functions in short intervals in the mean with respect to progressions, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 2, 341–362, 447 (Russian); English transl., Math. USSR-Izv. 30 (1988), no. 2, 315–335. MR 897001, DOI 10.1070/IM1988v030n02ABEH001013
- J. Wu, Théorèmes généralisés de Bombieri-Vinogradov dans les petits intervalles, Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 173, 109–128 (French). MR 1206205, DOI 10.1093/qmath/44.1.109
Additional Information
- Zhiwei Wang
- Affiliation: Institut Élie Cartan, Université de Lorraine, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France
- Email: zhiwei.wang@univ-lorraine.fr
- Received by editor(s): May 13, 2016
- Received by editor(s) in revised form: August 29, 2016
- Published electronically: January 31, 2017
- Additional Notes: The author was supported by the China Scholarship Council
- Communicated by: Matthew A. Papanikolas
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3211-3220
- MSC (2010): Primary 11N36, 11-XX, 11Nxx
- DOI: https://doi.org/10.1090/proc/13459
- MathSciNet review: 3652777