Zeta Functions on the Unitary Sphere

Abstract

In an earlier paper [5], the author defined a zeta function on the real sphere , whereas in the present paper it is proposed to define one on the unitary sphere where x_i's are complex numbers and their complex conjugates. Following E. Cartan, harmonics on the unitary sphere are defined and then a zeta function formed just as in the case of a real sphere. The unitary sphere is seen to behave like an even-dimensional closed manifold, since results similar to the ones proved by the author and A. Pleijel [6] for closed manifolds (of even dimensions) are observed here also.