Keating and Snaith have a famous conjecture on the asymptotics of the
integral $\int_0^T |\zeta(\frac 12+it)|^{2k}\, dt$, where $\zeta$
denotes the Riemann zeta function. See page 510 of the book review by
Brian Conrey of H. Iwaniec, *Lectures on the Riemann zeta function* in
*Bull. Amer. Math. Soc.* **53** (2016), 507--512
(http://www.ams.org/journals/bull/2016-53-03/S0273-0979-2016-01525-4/S0273-0979-2016-01525-4.pdf). This
conjecture involves a certain number $g_k$. This number is equal to
the number of standard Young tableaux whose shape is a $k\times k$
rectangle. Equivalently, this is the degree of the irreducible
representation of the symmetric group $S_{k^2}$ corresponding to this
shape. Is there any (conjectural) explanation for this connection
between $\zeta(s)$ and the representation theory of $S_{k^2}$? Does the
degree of other irreps of $S_n$ (for suitable $n$) also have
connections with the asymptotics of $\zeta(s)$?