CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

We prove that the unconditional asymptotic formula for the sum of the binomial coefficients ${\sum }_p\left(\genfrac{}{}{0.0pt}{}{n}{p}\right)$ over prime numbers $p\le n$ holds for almost all $n$. We also establish the bounds on this sum and show that improvements on the lower bound would imply a bound for prime gaps $g_n\ll \sqrt{p_n}$ which is stronger than Cramér’s bound $g_n\ll \sqrt{p_n}log{p}_n$ conditional on the Riemann Hypothesis.

This is a joint work with Nilotpal Kanti Sinha.