Combinatorics of lower order terms in the moment conjectures for the Riemann zeta function

Abstract

Conrey, Farmer, Keating, Rubinstein and Snaith have given a recipe that conjecturally produces, among others, the full moment polynomial for the Riemann zeta function. The leading term of this polynomial is given as a product of a factor explained by arithmetic and a factor explained by combinatorics (or, alternatively, random matrices). We explain how the lower order terms arise, and clarify the dependency of each factor on the exponent $k$ that is considered. We use extensively the theory of symmetric functions and representations of symmetric groups, ideas of Lascoux on manipulations of alphabets, and a key lemma, due in a basic version to Bump and Gamburd. Our main result ends up involving dimensions of skew partitions, as studied by Olshanski, Regev, Vershik, Ivanov and others. In this article, we also lay the groundwork for later unification of the combinatorial computations for lower order terms in the moments conjectures across families of $L$-functions of unitary, orthogonal and symplectic types.