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In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated Eisenstein series. It is named for Leopold Kronecker.

First Kronecker limit formula

The (first) Kronecker limit formula states that

$$E(\tau,s) = {\pi\over s-1} + 2\pi(\gamma-\log(2)-\log(\sqrt{y}|\eta(\tau)|^2)) +O(s-1)$$

where

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

$$E(\tau,s) =\sum_{(m,n)\ne (0,0)}{y^s\over|m\tau+n|^{2s}}$$

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.
$$\eta(\tau) = q^{1/24}\prod_{n\ge 1}(1-q^n),$$ with $$q = e^{2\pi i \tau }$$ 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.
Second Kronecker limit formula

The second Kronecker limit formula states that

$$E_{u,v}(\tau,1) = -2\pi\log|f(u-v\tau;\tau)q^{v^2/2}|$$

where

u and v are real and not both integers.
$$q = e^{2\pi i \tau }$$ and $$qa = e^{2\pi i a\tau }$$
$$p = e^{2\pi i z }$$ and $$pa = e^{2\pi i az }$$
$$E_{u,v}(\tau,s) =\sum_{(m,n)\ne (0,0)}e^{2\pi i (mu+n\tau)}{y^s\over|m\tau+n|^{2s}}$$

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

$$f(z,\tau) = q^{1/12}(p^{1/2}-p^{-1/2})\prod_{n\ge1}(1-q^np)(1-q^n/p).$$

Herglotz–Zagier function

References

Serge Lang, Elliptic functions, ISBN 0-387-96508-4
C. L. Siegel, Lectures on advanced analytic number theory, Tata institute 1961.

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