I do protest again against this put on hold ! It is an interesting problem with long history! It was first proposed at MGU Olympiad in 1976. It is also in deep connection with famous unusual Ramanujan's inequality and further Szego results.

May you give references to "Welch, Kabatianski, Levenshtein, Sidelnikov", please.

Due to information in 2013 V.P.Zastavny thesis a problem was first in more general form formulated in Schoenberg,1938,Tran.AMS 44. Interesting generalizations are in Zastavnyi V. P. Positive definite functions depending on the norm / V. P. Zastavnyi // Russian J. Math. Physics. – 1993. – Vol. 1,ð4. – P. 511– 522. and Koldobskii : mathnet.ru/php/…

Is it complete?

Here posted by "arqady" (Arqady Rosenberg): dxdy.ru/topic112404.html . No comments still! Difficuilt "schtuchka"!

This is a well-known problem, also posed on dxdy forum. Why to ban it immediately for no reason at all? Let start a company to ban a band of unprofessional banners!

The squared hypergeometric function under the integral is in fact the Legendre function, may be it will help?

@T.Amdeberhan - you are right with $\sqrt{\frac{5n}{3}}$, I mean arithmetic mean by mistake, thank you.

May be start with simple $(\sum x_k)^2=\sum x_k x_m$ -?

It is also possible to apply the Slater theorem - a general key to such integrals.

it is a derivative under the integral sigh?

May be it is true without integral? Try derivative?

sorry I was not accurate with scales.

thank you. Note on your graph for a point (1,1) it seems there are complex zeroes not on the cross of axes. But the next command in MATHEMATICA $NSolve[Cosh[z] Cos[z] + Sinh[z] Sin[z] == 0 && -100 <= Re[z] <= 100 && -100 <= Im[z] <= 100, z]$ gives all zeroes on the cross.

thank you for useful calculations. It is quiet possible that mine calculations were not accurate, I am not very good in them. But may you to plot $D$ in $(a,b)$ - plane approximately? And to prove strictly that $D$ is unbounded is also an interesting problem , not so?