Explicit Formulae

Abstract

This chapter discusses the approach to the superzeta functions (introduced in the previous chapter) by means of Explicit Formulae. In number theory, Explicit Formulae designate certain summation formulae connecting the primes and the Riemann zeros [55, Chap. IV]. As we may also refer to explicit formulae in their generic sense of closed-form results, we will specifically use capital initials to distinguish that number-theoretical meaning. We focus on just one type of Explicit Formula, the Guinand–Weil form [42, 43, 111], which accommodates general test functions and appears closest to the Poisson summation formula [50]. It is formally capable of evaluating superzeta functions, and we will try to use it for that goal.