On the number of irreducible representations of $\mathfrak{su}(3)$

In this note, we use a variant of the hyperbola method to prove an asymptotic expansion for the summatory function of the number of irreducible $\mathfrak{su}(3)$-representations of dimension $n$. This is a natural companion result to work of Romik, who proved an asymptotic formula for the number of unrestricted $\mathfrak{su}(3)$-representations of dimension $n$.


Introduction and statement of results
The irreducible representations of the Lie algebra su(3) are a family of representations W j,k of dimension jk(j+k) 2 for j, k ∈ N 0 (see [6,Theorem 6.27]).Let r(n) denote the number of n-dimensional representations of su (3).Then n≥0 r(n)q n = j,k≥1 1 1 − q jk(j+k) 2 . (1.1) Romik recently proved the following asymptotic formula for r(n) by studying (a renormalization of) the Witten zeta function for SU (3); that is, the meromorphic continuation of the series Theorem 1.1 ([9], Theorem 1.1).As n → ∞ , we have, for certain constants Romik stated that Theorem 1.1 is an analogue of the Hardy-Ramanujan asymptotic formula for p(n), the number of integer partitions of n, because the corresponding generating function for su(2)-representations coincides with (1. 2) The doubly indexed product (1.1) has much more complicated analytic behavior compared to the modular infinite product (1.2).Two of the authors [5] subsequently obtained an asymptotic series for r(n) which was then generalized by the authors and Brindle to more general product generating functions [4], including for example representations of so( 5).
In the present paper, we turn our attention to the number of irreducible su(3)representations of dimension n, i.e., Of course, this is a highly oscillatory function and is often 0, but we may still study the average, Dirichlet's hyperbola method yields the first two terms in the expansion of the average, where γ is the Euler-Mascheroni constant (see for example [1,Theorem 3.3]).The still open Dirichlet divisor problem concerns improving the error term from O( √ x) to the conjectured O(x 1 4 ); for an overview, see [2].We show here that a variant of the hyperbola method yields the following asymptotic expansion for the summatory function of ̺(n).
We prove Theorem 1.2 in Section 2, and we conclude this section with the following questions.
(1) Can one improve the error term in Theorem 1.2, perhaps by a deeper study of the Witten zeta function, ζ su(3) (s)?It would be reasonable to consider deeper techniques that have been brought to bear on the Dirichlet divisor problem (the Selberg-Delange method [10], Voronoi summation [7], to name a few).( 2) Can this variant of the hyperbola method (or any other technique) be used to yield asymptotic series for generic sums where p(x, y) is a homogeneous polynomial in Q[x, y] taking integer values?For example, the case p(m, n) = mn(m+n)(m+2n) 6 corresponds to representations of so (5).

1.
Now mn(m + n) ≡ 0 (mod 2) is automatically satisfied, and we see that the above sum counts lattice points in the (m, n)-plane between m = 1, n = 1 and the curve (the positive solution to the quadratic equation mn(m + n) = 2x).In usual hyperbola-method fashion, we add up the lattice points for each 1 ≤ m ≤ x ⌋.By symmetry these are the same.Then we subtract the points counted twice, namely those in the square with side length Thus, we only need to approximate the remaining sum.Let {t} := t − ⌊t⌋.Abel partial summation [10, Theorem 0.3, p. 4] gives .
Remark.The proof of Theorem 1.2 implies the identity We did not find a direct proof, but we note that the factor appears in evaluations of the complete elliptic integral of the first kind [3, Table 9.1].