Kronecker Limit Formula - First Kronecker Limit Formula

First Kronecker Limit Formula

The (first) Kronecker limit formula states that

where

• E(τ,s) is the real analytic Eisenstein series, given by

for Re(s) > 1, and by analytic continuation for other values of the complex number s.

• γ is Euler-Mascheroni constant
• τ = x + iy with y > 0.
• , with q = e2π i τ is the Dedekind eta function.

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.