On the error term in Weyl's law for the Heisenberg manifolds

TL;DR

This paper investigates the error term in Weyl's law for Heisenberg manifolds, deriving asymptotic formulas for moments and establishing the distributional properties of a scaled version of the error term.

Contribution

It provides the first asymptotic formulas for the moments of the error term and proves the distributional behavior of a scaled error term in Heisenberg manifolds.

Findings

Asymptotic formulas for the k-th power moments of the error term for 3 ≤ k ≤ 9

Proof that the scaled error term t^{-(l-1/4)} R(t) has a distribution function

Extension of Weyl's law error analysis to higher moments in Heisenberg manifolds

Abstract

For a fixed integer $l\geq 1$, let $R(t)$ denote the error term in the Weyl's law of a $(2l+1)$-dimensional Heisenberg manifold with the metric $g_l.$ In this paper we shall prove the asymptotic formula of the $k$-th power moment for any integers $3\leq k\leq 9.$ We shall also prove that the function $t^{-(l-1/4)}R(t)$ has a distribution function.