An explicit version of Bombieri's log-free density estimate and Sárközy's theorem for shifted primes
Abstract
We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near $\sigma = 1$" for Dirichlet $L$-functions. We use this estimate and recent work of Green to prove that if $N\geq 2$ is an integer, $A\subseteq\{1,\ldots,N\}$, and for all primes $p$ no two elements in $A$ differ by $p-1$, then $|A|\ll N^{1-1/10^{18}}$. This strengthens a theorem of Sárközy.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2022
- DOI:
- 10.48550/arXiv.2208.11123
- arXiv:
- arXiv:2208.11123
- Bibcode:
- 2022arXiv220811123T
- Keywords:
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- Mathematics - Number Theory
- E-Print:
- 24 pages. v4: Incorporates referee comments