The following is a piece of math lore from the late forties, which was described in an Intelligencer article entitled "The Elementary Proof of the Prime Number Theorem". It reads:
Turán, who was eager to catch up with the mathematical developments that had occurred during the war, talked with Selberg about his sieve method and his now famous inequality.He tried to talk Selberg into providing a seminar, showing the power of his inequality by giving an elementary proof of Dirichlet’s Theorem on primes in arithmetic progressions; but Selberg, who was busy with other research and was also looking for a permanent academic position, declined. He suggested that Turán present the seminar, using the notes he had made for himself from his conversations with Selberg.
Turán went through with this and afterwards, much to everyone's apparent incredulity, Erdős remarked "I think you can also derive $\frac{p_{n+1}}{p_n} \to 1$, referring to the aforementioned inequality of Selberg." (And, lo and behold, Erdős was able to do just that)
Two questions: (1) what inequality exactly is being referred to here?, and (2) how is Erdős's result deduced?
ADDENDUM: Here was the reason for my confusion. The formula (Selberg's identity) appears very early on in the paper. Then, several pages later we have "...talked with Selberg about his sieve method and now famous inequality" and then in the next paragraph Erdős claims to be able to derive the result from "the inequality". This suggested to me that the referenced inequality had nothing to do with the first identity, but it was instead some well-known sieve-theoretic namesake of Selberg. (Forgive my complete lack of knowledge of sieve theory, but it seemed as if there was nothing sieve-like about the identity, which is why I did not make the connection.)