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Statement

Notation (p. 251): $p_k$ is the $k$th prime.

The question (p. 256), restated. The paper asks whether the lower density ("untere Dichte") of the set of $k$ with

$$\frac{p_k}{k}<\frac{p_{k+1}}{k+1},$$

respectively of the set of $k$ with $p_k/k>p_{k+1}/(k+1)$, is positive.

The second set (p. 256). For the $k$ with $p_k/k>p_{k+1}/(k+1)$ the paper gives a short argument that the answer is yes. This inequality is equivalent to $k(p_{k+1}-p_k)<p_k$. As in the proof of Satz 1, the $p_k$ with $p_{k+1}-p_k<(1-\delta)\log k$ have positive density when $\delta>0$ is chosen small enough (independently of $k$); since $p_k>(1-\delta)k\log k$ for all sufficiently large $k$, each such $k$ has $k(p_{k+1}-p_k)<p_k$.

The first set (p. 256). The paper closes the paragraph with the remark that it "scheint schwierig zu sein, zu beweisen, daß die $k$ mit $\frac{p_k}{k}<\frac{p_{k+1}}{k+1}$ positive untere Dichte haben" (it seems difficult to prove that the $k$ with $p_k/k<p_{k+1}/(k+1)$ have positive lower density). The paper proves nothing about this set's density.

Other problems on the same pages (pp. 255--256), recorded for completeness. The paper conjectures that for each $\varepsilon$ there is an $l=l(\varepsilon)$ such that, for all $p_k<x$ except $\varepsilon x/\log x$ values of $k$, $p_k/k<\max_{1\le i\le l}p_{k+i}/(k+i)$ (its (15), p. 255); it conjectures that no $k$, or only finitely many, satisfy $\max_{1\le i<k}p_{k-i}/(k-i)<p_k/k<\min_{1\le i<\infty}p_{k+i}/(k+i)$ (p. 256); and it asks whether $p_k/k>p_{k+1}/(k+1)>p_{k+2}/(k+2)$ can occur infinitely often (p. 256).

Source. P. Erdős and K. Prachar, Sätze und Probleme über $p_k/k$, Abh. Math. Sem. Univ. Hamburg 25 (1961/1962), 251--256, doi:10.1007/BF02992930; the question, the argument for the second set and the remark on the first set on p. 256, conjecture (15) on p. 255. The edition read is identified on the source card.

Read depth. Claims checked: the question, the argument for the second set and the remark on the first were read clause by clause on the print. Nothing here is independently reviewed.

Dependencies

The short-gap count from the proof of Satz 1 and the prime number theorem.

Bears on

• Problem 968: the problem asks whether the set of $n$ with $p_n/n<p_{n+1}/(n+1)$ has positive density, and its precise statement asks for positive lower density, which is this question for the first set. The paper answers the companion question for the set with $p_k/k>p_{k+1}/(k+1)$ and leaves the first set open, remarking only that it seems difficult.