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I was looking at the OEIS on the number of square Young Tableaux.

In it Michael Somos referenced a paper of Conrey and Gonek, High Moment's of the Riemann Zeta-Function. Is there an combinatorial interpretation of equation $(19)$ on page 8 in the paper?

I'm trying to piece one together, but I have a minuscule knowledge about number theory. The coefficient $\frac{g_k}{k^2!}$ is the probability that a $k$ by $k$ grid of $k^2$ distinct numbers is a young tableaux. The probability that $k^2$ numbers of size $N$ are all prime is $\log^{-k^2}(N)$, so it sort of looks like a conditional probability? Is there a simple interpretation of what $a_k$ (equation 10 on page $4$) means?

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    $\begingroup$ Can you please state your question in a self-contained way, without making people read stuff from other links? $\endgroup$
    – Sam Hopkins
    Commented Dec 11, 2021 at 21:53
  • $\begingroup$ A related question is mathoverflow.net/questions/248784. $\endgroup$
    – Richard Stanley
    Commented Dec 11, 2021 at 23:04

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