So the Eisenstein series has a pole at s = 1 of residue π, and the (first) Kronecker limit formula gives the constant term of the
Laurent series at this pole.
This formula has an interpretation in terms of the
spectral geometry of the elliptic curve associated to the lattice : it says that the
zeta-regularized determinant of the Laplace operator associated to the flat metric on is given by . This formula has been used in
string theory for the one-loop computation in
Polyakov's perturbative approach.
Second Kronecker limit formula
The second Kronecker limit formula states that
where
u and v are real and not both integers.
q = e2π i τ and qa = e2π i aτ
p = e2Ï€ i z and pa = e2Ï€ i az
for Re(s) > 1, and is defined by analytic continuation for other values of the complex number s.