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January 2006 Primes in tuples III: On the difference {$p_{n + \nu}- p_n$}
Daniel Goldston, János Pintz, Cem Yalç cı m Yıldırım
Funct. Approx. Comment. Math. 35: 79-89 (January 2006). DOI: 10.7169/facm/1229442618


In the present work we prove a new estimate for $\Delta_\nu:=\liminf_{n \to \infty} \frac{(p_{n+\nu}-p_n)}{\log p_n}$, where $p_n$ denotes the $n$th prime. Combining our recent method which led to $\Delta_1=0$ with Maier's matrix method, we show that $\Delta_\nu\leq e^{-\gamma}(\sqrt{\nu}-1)^2$. We also extend the result to primes in arithmetic perogressions where the modulus can tend slowly to infinity as a function of $p_n$.


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Daniel Goldston. János Pintz. Cem Yalç cı m Yıldırım. "Primes in tuples III: On the difference {$p_{n + \nu}- p_n$}." Funct. Approx. Comment. Math. 35 79 - 89, January 2006.


Published: January 2006
First available in Project Euclid: 16 December 2008

zbMATH: 1196.11123
MathSciNet: MR2271608
Digital Object Identifier: 10.7169/facm/1229442618

Primary: 11P32
Secondary: 11N36

Keywords: prime numbers

Rights: Copyright © 2006 Adam Mickiewicz University

Vol.35 • January 2006
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