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Claim. Theorem 2 of D. A. Goldston, J. Pintz and C. Y. Yıldırım, Primes in tuples I, Ann. of Math. (2) 170 (2009), no. 2, 819--862, states

$$\liminf_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=0 .$$

The paper is described on its library card. Choosing a strictly increasing sequence $n_i$ along which the ratio tends to $0$, and using $\log p_n\sim\log n$, gives $(p_{n_i+1}-p_{n_i})/\log n_i\to0$: the case $C=0$ of Problem 5, answered yes. The proof weights admissible tuples with a truncated divisor sum and uses the Bombieri–Vinogradov theorem as its level of distribution.

Covers. The case $C=0$.

Depends on. Nothing in this wiki.

Acceptance. Refereed publication: Ann. of Math. (2) 170 (2009), no. 2, 819--862, doi:10.4007/annals.2009.170.819; the arXiv version was first posted on 10 August 2005, the date of this page. Not reviewed: the site's commentary credits $0$ as a limit point to [GPY09], but the site labels the problem OPEN, so that commentary is not acceptance.