mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Chowla–Selberg formula is the evaluation of a certain product of values of the

gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

at rational values in terms of values 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 ...

at imaginary quadratic irrational numbers. The result was essentially found by and rediscovered by .

Statement

In logarithmic form, the Chowla–Selberg formula states that in certain cases the sum :

\frac\sum_r \chi(r)\log \Gamma\left( \frac \right) = \frac\log(4\pi\sqrt)
+\sum_\tau\log\left(\sqrt, \eta(\tau), ^2\right)

can be evaluated using the

Kronecker limit formula 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 com ...

. Here χ is the quadratic residue symbol modulo ''D'', where ''−D'' is the

discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...

of an imaginary

quadratic field In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...

. The sum is taken over 0 < ''r'' < ''D'', with the usual convention χ(''r'') = 0 if ''r'' and ''D'' have a common factor. The function η is the

, and ''h'' is the class number, and ''w'' is the number of roots of unity.

Origin and applications

The origin of such formulae is now seen to be in the theory of

complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...

, and in particular in the theory of periods of an

abelian variety of CM-type In mathematics, an abelian variety ''A'' defined over a field ''K'' is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(''A''). The terminology here is from complex multiplication theory, which was deve ...

. This has led to much research and generalization. In particular there is an analog of the Chowla–Selberg formula for

p-adic number In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infin ...

s, involving a

p-adic gamma function In mathematics, the ''p''-adic gamma function Γ''p'' is a function of a ''p''-adic variable analogous to the gamma function. It was first explicitly defined by , though pointed out that implicitly used the same function. defined a ''p''-adic ...

, called the

Gross–Koblitz formula In mathematics, the Gross–Koblitz formula, introduced by expresses a Gauss sum using a product of values of the ''p''-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Dav ...

. The Chowla–Selberg formula gives a formula for a finite product of values of the eta functions. By combining this with the theory of

, one can give a formula for the individual absolute values of the eta function as :

\Im(\tau), \eta(\tau), ^4 = \frac \prod_r\Gamma(r/, D, )^

for some algebraic number α.

Examples

Using

Euler's reflection formula In mathematics, a reflection formula or reflection relation for a function is a relationship between and . It is a special case of a functional equation. It is common in mathematical literature to use the term "functional equation" for what a ...

for the gamma function gives: *

\eta(i) = 2^\pi^\Gamma(\tfrac)

References

* * * * {{DEFAULTSORT:Chowla-Selberg formula Theorems in number theory Gamma and related functions

Statement

Origin and applications

Examples

See also

References