We can define an integral of a step function against φ as The definition extends to more general projective systems, such as those indexed by the positive integers ordered by divisibility.
As an important special case consider the projective system Z/nZ indexed by positive integers ordered by divisibility.
The multiplication theorem for the Hurwitz zeta function gives a distribution relation Hence for given s, the map
Consider any eigenfunction of the Hecke operator Tp with eigenvalue λp prime to p. We describe a procedure for deriving a measure of ZD.
Fix an integer N prime to p and to D. Let F be the D-module of all functions on rational numbers with denominator coprime to N. For any prime l not dividing N we define the Hecke operator Tl by Let f be an eigenfunction for Tp with eigenvalue λp in D. The quadratic equation X2 − λpX + p = 0 has roots π1, π2 with π1 a unit and π2 divisible by p. Define a sequence a0 = 2, a1 = π1+π2 = λp and so that