Erdos 1961 satze und probleme uber german

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question_p256: Erdős and Prachar's question whether the k with p_k/k < p_{k+1}/(k+1), and the k with p_k/k > p_{k+1}/(k+1), have positive lower density, with their argument for the second set and their remark that the first seems hard; the question of Problem 968.

satz_1: Erdős and Prachar's two-sided bound c_1 log^2 x < sum over p_k <= x of |p_{k+1}/(k+1) - p_k/k| < c_2 log^2 x for suitable positive constants, with the consequence that p_k/k is not monotone from any point on.

satz_2: Erdős and Prachar's bound for a subsequence p_{k_i} of the primes with p_{k_i}/k_i < p_{k_{i+1}}/k_{i+1} for every i: its terms up to x number o(x/log x); the closing remark (p. 256) says the same method gives O(x/log^{1+delta} x) for sufficiently small delta, for instance any delta < 1/4.

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P. Erdős, K. Prachar: Sätze und Probleme über $p_k/k$ (in German), Abh. Math. Sem. Univ. Hamburg 25 (1961/1962), 251--256; MR 25 #3901; Zentralblatt 107,266.

Writing p_k for the k-th prime, the paper studies the fluctuation of the sequence p_k/k, which by the prime number theorem is asymptotic to log k. Satz 1 proves two-sided bounds c_1 log^2 x < sum_{p_k <= x} |p_{k+1}/(k+1) - p_k/k| < c_2 log^2 x for suitable positive constants, and the authors point out the immediate consequence that p_k/k cannot be monotone from any point onwards. Satz 2 shows that any subsequence p_{k_i} along which p_{k_i}/k_i < p_{k_{i+1}}/k_{i+1} holds has only o(x / log x) terms up to x, and the closing remark (p. 256) says the same method gives O(x / log^{1+delta} x) for small delta, for instance any delta < 1/4. The proof of Satz 1 counts primes with prescribed gap sizes, combining the prime number theorem with a Brun-sieve/Schnirelman estimate for the number of p_k with p_{k+1} - p_k equal to a fixed value n, bounded by c_4 (x/log^2 x) sum_{d | n} 1/d. The paper also poses further problems about p_k/k. On p. 256 it asks whether the k with p_k/k < p_{k+1}/(k+1), and the k with p_k/k > p_{k+1}/(k+1), have positive lower density; it shows that the second set has positive density and says that proving positive lower density for the first set seems difficult. The question for the first set is the precise statement of Problem 968.

Source: https://users.renyi.hu/~p_erdos/1961-21.pdf. No copyright or license line is printed on the pages; the publisher's article page was not consulted, and the Crossref record for DOI 10.1007/bf02992930 (read 2026-10-02) names only Springer's text-and-data-mining terms (http://www.springer.com/tdm) and no Creative Commons license, every other right reserved.

Bears on. #968: the question on p. 256 asks whether the k with p_k/k < p_{k+1}/(k+1) have positive lower density, which is the problem's precise statement. The paper shows that the k with p_k/k > p_{k+1}/(k+1) have positive density, says that the first set seems difficult, and proves nothing about it.

Results.

• Satz 1 (p. 251; proof pp. 251--253): there are constants c_1, c_2 > 0 with c_1 log^2 x < sum_{p_k <= x} |p_{k+1}/(k+1) - p_k/k| < c_2 log^2 x; in particular p_k/k is not monotone from any point on.
• Satz 2 (p. 251; proof pp. 253--255): if p_{k_i} is a subsequence of the primes with p_{k_i}/k_i < p_{k_{i+1}}/k_{i+1} for all i, then the number of such p_{k_i} up to x is o(x / log x); the closing remark (p. 256) says the same method gives O(x / log^{1+delta} x) for sufficiently small delta, for instance any delta < 1/4.
• The lower-density question (p. 256): whether the k with p_k/k < p_{k+1}/(k+1), and the k with p_k/k > p_{k+1}/(k+1), have positive lower density, with the argument for the second set; the page also records the paper's other conjectures and questions on pp. 255--256.

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