We start from a known formula and set 5s instead of s:

Make the substitution x=2^t

now we multiply 2^t up and down.... and define

Now we make use of the identity:

(1)

for n big but finite (1) is an aproximation ot dirac,s delta function but (1) is in L² (1,∞) function space so we have

where it can be proved that K(s,t)=K(t,s) so if the kernel is symmetric their Eigenfunctions are orfthogonal and their eigenvalues are real so we can solve the integral equation for Pi(x) by :

now we only have to set t=log(x) log in basis 2and g(t) was by definition Pi(2^t)/2^2t