Young-lattice diagonals and a doubly graded multiple-zeta decomposition of $e^{\gamma }$

Abstract

An equivalent formulation of the Riemann hypothesis recently led to a partition expansion naturally indexed by diagonals of the Young lattice. Segovia isolated the hook families $(r,1^m)$ on these diagonals and computed their limiting contributions ${\rho }_r$, while observing that non-hook families provide a missing contribution. We introduce a bivariate finite generating function that packages all Young shapes on every fixed-excess diagonal at once. For each fixed $r\ge 1$, we obtain a diagonal generating polynomial $D_r(n;z)$ and prove \[ A_r(n)\sim C_r\,n\log\log n, \] where $C_r$ is the $(r-1)$st coefficient of an explicit convergent infinite product. Moreover, \[ C_r=\sum_{ν\vdash r-1} C_ν, \] giving a canonical decomposition over the partitions of the excess $r-1$. The one-part contribution is Segovia's hook constant ${\rho }_r$, while the remaining terms give all non-hook corrections simultaneously. We then refine these constants by introducing coefficients $C_{r,d}$ that record simultaneously the Young-lattice excess $r-1$ and the number $d$ of non-unit rows. Row sums recover the fixed-excess constants $C_r$, while column sums recover the depth decomposition in an Abel-regularized multiple-zeta expansion of $e^{\gamma }$. More precisely, each partition $\nu ⊢r-1$ is identified with an Abel-regularized multiple-zeta block of depth $\ell (\nu )$. Thus the same array $(C_{r,d})$ organizes the decomposition simultaneously by Young-lattice excess and multiple-zeta depth. Our results concern the combinatorial and asymptotic structure of this decomposition, rather than the Riemann hypothesis itself.

33 pages, 1 figure