In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series

Real analytic Eisenstein series

In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL and in analytic number theory...

 (or Epstein zeta function) in terms of the Dedekind eta function

Dedekind eta function

The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive...

. There are many generalizations of it to more complicated Eisenstein series. It is named for Leopold Kronecker

Leopold Kronecker

Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...

.

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
  Euler-Mascheroni constant
  The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
  
  
• τ = x + iy with y > 0.
• , with q = e^{2π i τ} is the Dedekind eta function
  Dedekind eta function
  The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive...
  
  .

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

Laurent series

In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...

at this pole.

Second Kronecker limit formula

The second Kronecker limit formula states that


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}
• 

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

•