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Statement

Setting (pp. 371--372). $p_1=2,p_2=3,\ldots$ are the primes. The paper asks whether, for every $t$, both inequalities

$$\Bigl(\frac{p_{n-1}^t+p_{n+1}^t}{2}\Bigr)^{1/t}>p_n \qquad (3)$$ $$\Bigl(\frac{p_{m-1}^t+p_{m+1}^t}{2}\Bigr)^{1/t}<p_m \qquad (4)$$

have infinitely many solutions ($n$ in (3), $m$ in (4)).

Theorem 1 (p. 372, quoted). "The inequalities (3) and (4) have infinitely many solutions."

Special cases (pp. 371--372). The paper's (1) is the pair $p_{n-1}p_{n+1}>p_n^2$, $p_{m-1}p_{m+1}<p_m^2$, the case $t=0$ of (3) and (4) when the power mean at $t=0$ is read as the geometric mean (the reading the paper uses on p. 375); its (2) is the pair $(p_{n-1}+p_{n+1})/2>p_n$, $(p_{m-1}+p_{m+1})/2<p_m$, the case $t=1$. So $\log p_n$ is not convex for all large $n$, which answers the question that opens the paper, and the primes are neither convex nor concave from any point on. The paper notes that the first inequality of (2) already follows from $\limsup(p_{n+1}-p_n)=\infty$ (p. 371), and that, since $((a^t+b^t)/2)^{1/t}$ increases with $t$, (1) and (2) follow from (3) and (4) and it suffices to prove (3) for $t<0$ and (4) for $t>0$ (p. 372).

The only fact about primes used (p. 372) is the Chebyshev-type bound $\pi(x)>c_1x/\log x$, the paper's (5).

Read depth. Claims checked: the statement, the questions (3) and (4), the special cases and the reduction were read clause by clause on the page images of pp. 371--374 of the print, and the proof on pp. 373--374 was followed for structure. Nothing here is independently reviewed.

Proof pointer

Pp. 373--374. By monotonicity of the power mean in $t$ it suffices to take $t=-l$ with $l\ge2$ an integer for (3), and an integer $t=l\ge2$ for (4). For (3) take $k$ from the Lemma's inequalities (6) with $A<1/(2l^2)$, so the gap $u=p_k-p_{k-1}$ is smaller than the next gap and below $p_k^{1/2}/(2l^2)$; reducing to the case $p_{k+1}-p_k=u+1$, a binomial expansion of $(p_k-u)^l$ and $(p_k+u+1)^l$ gives (3). For (4) the inequalities (7) of the Lemma are used in the same way. The Remark on p. 374 says Theorem 1 also follows from (5), the Lemma and Theorems 2 and 3. Section 3 (pp. 375--377) gives a second proof of the $t=1$ case (2), using Page's form of the prime number theorem for arithmetic progressions and a result of Kuzmin on exponential sums.

Dependencies

• Lemma (p. 372).
• The bound $\pi(x)>c_1x/\log x$, which the paper takes from the first pages of Ingham's The distribution of prime numbers.

Source. P. Erdős and P. Turán, On some new questions on the distribution of prime numbers, Bull. Amer. Math. Soc. 54 (1948), 371--378; the edition read is named on the source card.

Bears on

• Problem 6, as context only: the case $t=1$ of (3), $p_{n+1}-p_n>p_n-p_{n-1}$ infinitely often, is the statement for two consecutive gaps; the problem asks for three consecutive increasing gaps, which the theorem does not give.