We prove that $$ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $$ where$p$_{$n$}denotes the$n$th prime. Since on average$p$_{$n$+1}−$p$_{$n$}is asymptotically log_{$n$}, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences$p$−$p$′ between primes which includes the small gap result above.

No abstract uploaded!

If$n$is a non-negative integer, then denote by ∂^{-$n$}$H$^{∞}the space of all complex-valued functions$f$defined on $\mathbb{D}$ such that$f$,$f$^{(1)},$f$^{(2)},...,$f$^{($n$)}belong to$H$^{∞}, with the norm $$\|f\|=\sum_{j=0}^{n}\frac{1}{j!}\|f^{(j)}\|_{\infty}.$$ We prove bounds on the solution in the corona problem for ∂^{-$n$}$H$^{∞}. As corollaries, we obtain estimates in the corona theorem also for some other subalgebras of the Hardy space$H$^{∞}.

No abstract uploaded!

No abstract uploaded!