A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds

Abstract

Abstract. This article provides an Omega-result for the remainder term in Weyl's law for the spectral counting function of certain rational (2ℓ + 1) -dimensional Heisenberg manifolds.

A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds, II

“…As already stated, the case of even ℓ has been treated in part I of this work [20]. Therefore, we may suppose throughout that ℓ is odd.…”

“…The O -term here is in fact O(1) : see [20], f. (5.9). Now recall that (h, k) ∈ D(U ) explicitly means that…”

“…See also the detailed discussion in part I of this work [20]. (4) Recall the usual Ω -notation: For real functions F and G > 0 , and * denoting either + or − , F (x) = Ω * (G(x)) means that lim sup( * F (x)/G(x)) > 0 , as x → ∞ .…”

“…1. The case of even ℓ has been treated in the first part of this work [20]. After approximating R(t) by a suitable trigonometric sum, the Dirichlet approximation theorem was applied to give all its terms the positive sign.…”

Sign changes of the error term in Weyl’s law for Heisenberg manifolds

“…holds true for any 3 ≤ k ≤ 9, where C k,l and η k > 0 are explicit constants. Recently, Nowak [23,24] proved that the estimate…”