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Statement

Notation (p. 251): $p_k$ is the $k$th prime. By the prime number theorem $p_k/k\sim\log k$ as $k\to\infty$, and the paper contrasts the sequence with the increasing function $\log k$, whose increments over $p_k\le x$ add up to asymptotically $\log x$.

Satz 1 (p. 251), restated. There are constants $c_1,c_2>0$ such that

$$c_1\log^2x<\sum_{p_k\le x}\left|\frac{p_{k+1}}{k+1}-\frac{p_k}{k}\right|<c_2\log^2x .$$

The print says "bei passenden $c_1, c_2 > 0$" (for suitable $c_1,c_2>0$) and names no range of $x$; the proof (pp. 251--253) obtains both bounds for all sufficiently large $x$.

Consequence (p. 251). The paper notes that it follows in particular that the sequence $p_k/k$ is not monotone from any point on.

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; Satz 1 on p. 251, its proof on pp. 251--253. The edition read is identified on the source card.

Read depth. Claims checked: the statement and its consequence were read clause by clause on the print. The proof was read for its structure, not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 251--253. For the lower bound the paper shows that, for a suitably small fixed $\delta>0$, more than $c_3x/\log x$ of the primes $p_k\in(x/2,x]$ have $p_{k+1}-p_k\le(1-\delta)\log x$. It bounds the number of primes in that range whose gap lies between $(1-\delta)\log x$ and $(1+\delta)\log x$ by the estimate, attributed to Schnirelman and obtained by Brun's method, that fewer than $c_4\,(x/\log^2x)\sum_{d\mid n}1/d$ primes $p_k$ have $p_{k+1}-p_k=n$; comparing with the total length of the gaps then forces many short gaps. Each short gap gives a decrease of $p_k/k$ of size at least about $(\delta-\varepsilon)\log k/(k+1)$, and summing these over the dyadic ranges of $(\sqrt x,x)$ gives the order $\log^2x$. For the upper bound the increases of $p_k/k$ add up to $O(\log x)$, while each decrease is less than $c_8\log k/k$, and $\sum_{k\le\pi(x)}\log k/k=O(\log^2x)$.

Dependencies

The prime number theorem and the Brun--Schnirelman upper bound for the number of primes with a prescribed gap, both quoted from the literature (pp. 251--252).

Bears on

• Problem 968: context only. The problem asks about the set of $k$ with $p_k/k<p_{k+1}/(k+1)$; Satz 1 measures the total size of the oscillation of $p_k/k$, and its statement gives no density for either the rising or the falling steps. The short-gap count in its proof is what the paper reuses on p. 256 for the falling steps; it says nothing about the rising steps.