Divisors of shifted primes
Number Theory
2011-01-11 v3
Abstract
We bound from below the number of shifted primes p+s<x that have a divisor in a given interval (y,z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine the number of shifted primes that appear in a multiplication table up to multiplicative constants.
Keywords
Cite
@article{arxiv.0905.0163,
title = {Divisors of shifted primes},
author = {Dimitris Koukoulopoulos},
journal= {arXiv preprint arXiv:0905.0163},
year = {2011}
}
Comments
33 pages. To appear in Int. Math. Res. Not. IMRN. Fixed a small mistake in the proof of Lemma 2.5 and made stylistic changes