Kronecker Limit Formula
   HOME

TheInfoList



OR:

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 In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
. There are many generalizations of it to more complicated Eisenstein series. It is named for
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, ...
.


First Kronecker limit formula

The (first) Kronecker limit formula states that :E(\tau,s) = + 2\pi(\gamma-\log(2)-\log(\sqrt, \eta(\tau), ^2)) +O(s-1), where *''E''(τ,''s'') is the real analytic Eisenstein series, given by :E(\tau,s) =\sum_ for Re(''s'') > 1, and by analytic continuation for other values of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
''s''. *γ is
Euler–Mascheroni constant Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
*τ = ''x'' + ''iy'' with ''y'' > 0. * \eta(\tau) = q^\prod_(1-q^n), with ''q'' = e2π i τ is the
Dedekind eta function In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
. 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 In mathematics, the Laurent series of a complex function f(z) 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 a Taylor series expansio ...
at this pole. This formula has an interpretation in terms of the spectral geometry of the elliptic curve E_\tau associated to the lattice \mathbb + \mathbb \tau: it says that the zeta-regularized determinant of the Laplace operator \Delta associated to the flat metric \frac , dz, ^2 on E_\tau is given by 4y , \eta(\tau), ^4. This formula has been used in
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
for the one-loop computation in Polyakov's perturbative approach.


Second Kronecker limit formula

The second Kronecker limit formula states that :E_(\tau,1) = -2\pi\log, f(u-v\tau;\tau)q^, , 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'' *E_(\tau,s) =\sum_e^ for Re(''s'') > 1, and is defined by analytic continuation for other values of the complex number ''s''. *f(z,\tau) = q^(p^-p^)\prod_(1-q^np)(1-q^n/p).


See also

* Herglotz–Zagier function


References

*
Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...
, ''Elliptic functions'', {{isbn, 0-387-96508-4 * C. L. Siegel
''Lectures on advanced analytic number theory''
Tata institute 1961.


External links


Chapter0.pdf
Theorems in analytic number theory Modular forms