Skip to main content Accessibility help
×
Home

POSITIVE PROPORTION OF SHORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF PRIMES

  • DANIELE MASTROSTEFANO (a1)

Abstract

We prove that for every $m\geq 0$ there exists an $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$ such that if $0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$ and $x$ is sufficiently large in terms of $m$ and $\unicode[STIX]{x1D706}$ , then

$$\begin{eqnarray}|\{n\leq x:|[n,n+\unicode[STIX]{x1D706}\log n]\cap \mathbb{P}|=m\}|\gg _{m,\unicode[STIX]{x1D706}}x.\end{eqnarray}$$
The value of $\unicode[STIX]{x1D700}(m)$ and the dependence of the implicit constant on $\unicode[STIX]{x1D706}$ and $m$ may be made explicit. This is an improvement of the author’s previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when allowing the parameters $m$ and $\unicode[STIX]{x1D706}$ to vary as functions of $x$ or replacing the set $\mathbb{P}$ of all primes by primes belonging to certain specific subsets.

Copyright

Footnotes

Hide All

The author is funded by a Departmental Award and by an EPSRC Doctoral Training Partnership Award.

Footnotes

References

Hide All
[1] Freiberg, T., ‘Short intervals with a given number of primes’, J. Number Theory 163 (2016), 159171.
[2] Mastrostefano, D., ‘Short intervals containing a prescribed number of primes’, J. Number Theory 197 (2019), 4961.
[3] Maynard, J., ‘Dense clusters of primes in subsets’, Compos. Math. 152(7) (2016), 15171554.
[4] Murty, M. R. and Murty, V. K., ‘A variant of the Bombieri–Vinogradov theorem’, in: Number Theory (Montreal, Quebec, 1985), CMS Conference Proceedings, 7 (American Mathematical Society, Providence, RI, 1987), 243272.
[5] Soundararajan, K., ‘The distribution of prime numbers’, in: Equidistribution in Number Theory, An Introduction, NATO Science Series II: Mathematics, Physics and Chemistry, 237 (eds. Granville, A. and Rudnick, Z.) (Springer, Dordrecht, 2007), 5983.
[6] Thorner, J., ‘Bounded gaps between primes in Chebotarev sets’, Res. Math. Sci. 1(4) (2014), 16 pages.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

POSITIVE PROPORTION OF SHORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF PRIMES

  • DANIELE MASTROSTEFANO (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