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Statement

Setting. $\pi(x)$ is the number of primes not exceeding $x$. The paper recalls (p. 291, citing Landau's Handbuch, Vol. 1, §58) that $\pi(2x)<2\pi(x)$ for all sufficiently large $x$.

Problem 1 (p. 291). Erdős asks whether

$$\pi(x+y)\le\pi(x)+\pi(y)\qquad(1)$$

holds, with no range on $x$ and $y$ stated. He records that Ungár has verified the inequality for $y\le41$, and that Hardy and Littlewood proved

$$\pi(x+y)-\pi(x)<cy/\log y\qquad(2)$$

for a constant $c$, deducing it by Brun's method.

The surrounding conjectures (p. 291). With $\rho(y)=\limsup_{x\to\infty}\,[\pi(x+y)-\pi(x)]$, Hardy and Littlewood conjecture $\rho(y)>y/\log y$, and perhaps $\pi(y)-\rho(y)\to\infty$ as $y\to\infty$. Erdős describes as very difficult, weaker than (1) and much stronger than (2), the conjecture that for each $\varepsilon>0$ there is $y_\varepsilon$ such that for $y>y_\varepsilon$ (quoted) "$\pi(x + y) - \pi(y) < (1 + \varepsilon)y/\log y$" [sic]; the left side is printed with $\pi(y)$ where the comparison with (1) and (2) suggests $\pi(x)$. He adds that $\rho(y)=1$ for all $y$ has not been disproved, and draws a consequence for gaps between primes if $\rho(y)>1$ for some $y$.

The paper poses (1) and does not resolve it.

Source. P. Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), 291--300; §A, Problem 1, p. 291. The edition read is identified on the source card.

Read depth. Claims checked: the problem was read clause by clause on the page images of the journal print. A question has no proof to check; (2) is cited from Hardy and Littlewood, not proved here.

Dependencies

None.

Bears on

• Problem 855: inequality (1) is the problem's inequality. The site's wording asks it for large $x$ and $y$; the paper states no range. The paper poses the question and does not resolve it.