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.
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