Prolate spheroidal wave functions, Sonine spaces, and the Riemann zeta function

In this paper we study underlying spaces associated with A. Connesʼ trace formula (see Connes (1999) [3], Li (2010) [14]). In particular the explicit formula in the theory of prime numbers is expressed as the trace of an operator acting on a Hilbert space, which is a direct sum of a Sonine space, the space of prolate spheroidal wave functions, and a variant of the space of prolate spheroidal wave functions. A formula is obtained for the orthogonal projection of the Hilbert space onto the Sonine space. A base is given for the variant space of the space of prolate spheroidal wave functions.