# Does a proof of Selberg's 3.2 inequality exist?

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 In his 1978 survey article Montgomery states that Selberg has an unpublished proof that shows $$\frac{3}{2} \pi \approx 4.71$$ can be replaced by $$3.2$$.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

• 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 Commented Jun 13, 2013 at 22:57
• The quote in full ("Selberg (unpublished) has shown that 3$\pi$/2 can be replaced by 3.2, but it is not known whether the above holds with the constant $\pi$.")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. Commented Jun 13, 2013 at 23:08
• 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. Commented Jun 13, 2013 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. Commented Aug 18, 2013 at 3:13
• Incidentally, although it's nothing like 3.2, one can optimize Preissmann's proof a bit and obtain a bound $\approx 1.28\pi$, which is faintly better than Preissmann's $\pi(1+\frac{2}{3}(\frac{6}{5})^{1/2})^{1/2}\approx 1.32\pi$, by estimating the norm of $[\frac{w_r^{4/3}w_s^{2/3}}{(\lambda_r-\lambda_s)^2}]_{r,s}$ on $\ell_3$. Commented Mar 14, 2023 at 1:40