Kronecker limit formula

The (first) Kronecker limit formula states that

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

where

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

${\displaystyle E(\tau ,s)=\sum _{(m,n)\neq (0,0)}{y^{s} \over |m\tau +n|^{2s}}}$

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.
• ${\displaystyle \eta (\tau )=q^{1/24}\prod _{n\geq 1}(1-q^{n})}$, with q = e^{2π 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.

This formula has an interpretation in terms of the spectral geometry of the elliptic curve ${\displaystyle E_{\tau }}$ associated to the lattice ${\displaystyle \mathbb {Z} +\mathbb {Z} \tau }$: it says that the zeta-regularized determinant of the Laplace operator ${\displaystyle \Delta }$ associated to the flat metric ${\displaystyle {\frac {1}{y}}|dz|^{2}}$ on ${\displaystyle E_{\tau }}$ is given by ${\displaystyle 4y|\eta (\tau )|^{4}}$. This formula has been used in string theory for the one-loop computation in Polyakov's perturbative approach.

The second Kronecker limit formula states that

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

where

• u and v are real and not both integers.
• q = e^{2π i τ} and q^a = e^{2π i aτ}
• p = e^{2π i z} and p^a = e^{2π i az}
• ${\displaystyle E_{u,v}(\tau ,s)=\sum _{(m,n)\neq (0,0)}e^{2\pi i(mu+nv)}{y^{s} \over |m\tau +n|^{2s}}}$

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

• ${\displaystyle f(z,\tau )=q^{1/12}(p^{1/2}-p^{-1/2})\prod _{n\geq 1}(1-q^{n}p)(1-q^{n}/p).}$

• Herglotz–Zagier function

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

• Chapter0.pdf