- Kronecker limit formula
In mathematics, the classical Kronecker limit formula describes the constant term at "s" = 1 of a
real analytic Eisenstein series (orEpstein zeta function ) in terms of theDedekind eta function . There are many generalizations of it to more complicated Eisenstein series. It is named forLeopold Kronecker .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 isEuler-Mascheroni constant
*τ = "x" + "iy" with "y" > 0.
* , with "q" = e2π i τ is theDedekind 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.econd 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 "a"τ
*"p"=e2π i "z" and "pa"=e2π i "az"
*for Re("s") > 1, and is defined by analytic continuation for other values of the complex number "s".
*References
*
Serge Lang , "Elliptic functions", ISBN 0-387-96508-4
*C. L. Siegel , "Lectures on advanced analytic number theory", Tata institute 1961.External links
* [http://www.maths.warwick.ac.uk/~masfaw/Chapter0.pdf]
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