A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that
$$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r – \lambda_s} \right| \leq \pi \delta^{-1} \sum_{r} |w_{r}|^2$$
where $\{\lambda_{r}\}$ are an increasing sequence of $\delta$-separated real numbers ($|\lambda_{r+1} – \lambda_{r}| \geq \delta$) and $\{w_{r}\}$ are complex numbers. The constant $\pi$ is known to be sharp. There is a further `weighted' generalization of this inequality (also due to Montgomery and Vaughan) that states
$$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r – \lambda_s} \right| \leq \frac{3}{2} \pi \sum_{r} \delta_{r}^{-1}|w_{r}|^2$$
where $\delta_{r}>0$ is a real number such that $|\lambda_{r}-\lambda_{s}| \geq \delta_{r}$ for any $s \neq r$ and the rest of the notation is the same as above.
Here the constant $\frac{3}{2}\pi$ is not optimal, indeed it is conjectured that the $\frac{3}{2} \pi$ can be replaced by $\pi$. Proving this, however, has remained open for over 40 years. Refining the constant would have a number of applications to sieve theory (indeed this inequality even has a minor role in the ongoing Polymath project to refine Zhang's prime gap theorem). This article of Montgomery is a good place to read about the role of the inequality in number theory (as of 1978, at least).
There have been a number of refinements to the constant over the years. In his 1978 survey article Montgomery states that Selberg has an unpublished proof that shows $\frac{3}{2} \pi \approx 4.71$ can be replaced by $3.2$. Curiously, in 1984 E. Preissmann published a (18 page!) proof showing that the inequality holds with the constant $\frac{4}{3}\pi \approx 4.18$ (which is inferior to that claimed by Selberg). In addition, I have read that there is a proof of the inequality with constant $\sqrt{22} \approx 4.69$ given by Jörg Brüdern (in Einführung in die analytische Zahlentheorie, Springer Verlag 1995), which would be yet inferior to Preissmann's result. This leads me to ask:
Does there exist a copy of Selberg's proof?
Of course, I'd be interested to know of any results related to the problem beyond those listed above.
Best Answer
I think it is highly unlikely that such a proof exists. This is quite far from definitive, but that's usually the case with this kind of nonexistance answers. My reasoning is:
He was a very prolific writer (after WWII, anyway). His collected papers add up to more than 900 pages. As original a mathematician as Selberg was, I can't imagine him dismissing this result as "not important enought to publish".
The survey is from 1978, so Selberg (professor at Princeton until 1987) had more than enough time to publish such a relevant proof.
Finally, as Lucia mentions on the comments, a lot of work has gone into cataloguing and publishing his research since he retired. The first edition of the Springer volumes dates from 1989 and 1991, the newest from 2014. Springer is known to sometimes include things like unpublished papers to letters is his Collected Papers series, so if the proof was ever written down, it should have surfaced by now.