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 $p_n$.

Locations

• Functiones et Approximatio Commentarii Mathematici - View - PDF