Lower Order Terms in Szegö Type Limit Theorems on Zoll Manifolds†

Abstract

We compute 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. In Guillemin and Okikiolu (Guillemin, V., Okikiolu, K. ([1997a]). Spectral asymptotics of Toeplitz operators on Zoll manifolds. J. Funct. Anal. 146:496–516), the second order term was computed by Guillemin and Okikiolu. In the present article, an important role is played by a certain combinatorial identity which we call the generalized Hunt–Dyson formula (Gioev, D. ([2002b]). Generalized Hunt–Dyson formula and Bohnenblust–Spitzer theorem. Int. Math. Res. Not. 2002(32):1703–1722). 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 of a Toeplitz matrix T 1 with the product of another Toeplitz matrix T 2 and a diagonal matrix D of the form . Here , n ones. †Dedicated to the memory of my grandparents, Valentina Nikolaevna Pavlova and Dimitri Aleksandrovich Golubentsev.