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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 In his 1978 survey article Montgomery states that Selberg has an unpublished proof that shows $\frac{3}{2} \pi \approx 4.71$ can be replaced by $3.2$.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.

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How sure are you that 3.2 isn't a typo for 4.2? Gerhard "Just A Finger Width's Difference" Paseman, 2013.06.13 –  Gerhard Paseman Jun 13 '13 at 22:57
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The quote in full ("Selberg (unpublished) has shown that 3$\pi$/2 can be replaced by 3.2, but it is not known whether the above holds with the constant $\pi$.") makes more sense with 3.2 than 4.2. –  Sam Hopkins Jun 13 '13 at 23:08
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Montgomery writes in his 1994 "Ten Lectures..." book: "Preissmann [51] has replaced 3/2 by a constant a littler smaller than 4/3, and Selberg (unpublished) claims to have proved (27) with the constant 3.2." If 3.2 is a typo its a persistent one. On the other hand, Montgomery's phrasing is a bit more equivocal here. –  Mark Lewko Jun 13 '13 at 23:35
    
Some of Selberg's unpublished papers are now available at publications.ias.edu/selberg and the full collection is cataloged here library.ias.edu/finding-aids/selberg#ref132 However I don't know if this result appears in any of these papers. –  Lucia Aug 18 '13 at 3:13
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