## Functiones et Approximatio Commentarii Mathematici

### Primes in tuples III: On the difference {$p_{n + \nu}- p_n$}

#### Abstract

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 $p_n$.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 35 (2006), 79-89.

Dates
First available in Project Euclid: 16 December 2008

https://projecteuclid.org/euclid.facm/1229442618

Digital Object Identifier
doi:10.7169/facm/1229442618

Mathematical Reviews number (MathSciNet)
MR2271608

Zentralblatt MATH identifier
1196.11123

Keywords
prime numbers

#### Citation

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