Lower Order Terms in Szegö Theorems on Zoll Manifolds

We give an outline of the computation of the third order term in a generalization of the Strong Szegö Limit Theorem for a zeroth order pseudodifferential operator (PsDO) on a Zoll manifold of an arbitrary dimension, see [Gi2] for the detailed proof. This is a refinement of a result by V. Guillemin and K. Okikiolu who have computed the second order term in [GO2]. An important role in our proof is played by a certain combinatorial identity which generalizes the formula of G. A. Hunt and F. J. Dyson to an arbitrary natural power, see [Gi3]. This identity is a different form of the renowned Bohnenblust–Spitzer combinatorial theorem which is related to the maximum of a random walk with i.i.d. steps on the real line. A corollary of our main result is a fourth order Szegö type asymptotics for a zeroth order PsDO on the unit circle, which in matrix terms gives a fourth order asymptotic formula for the determinant of the truncated sum Pn(T1 +T2D)Pn of a Toeplitz matrix T1 with the product of another Toeplitz matrix T2 and a diagonal matrix D of the form diag(· · · , 1 3 , 1 2 , 1, 0, 1, 1 2 , 1 3 , · · · ). Here Pn = diag(· · · , 0, 1, · · · , 1, 0, · · · ), (2n + 1) ones.