Problem 15

Erdős Problem #15: declared status 'open'. Formalized: yes. Is it true that$$\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}$$converges, where $p_n$ is the sequence of primes? Current best: Sawhney has provided the following proof that this series converges absolutely for $c>2$: note that, whenever $c>1$, the contribution to the sum from gaps $p_{n+1}-p_n\geq \log n$ is convergent, so it suffices to consider only small gaps. The number of $n\leq X$ such that $p_{n+1}-p_n\in [\epsilon\log n,2\epsilon \log n)$ is bounded above by $\ll \epsilon X$ (this can be proved via the Selberg sieve). Prize: no. Tags: number theory, primes.