Functional equation (L-function)
Encyclopedia
In mathematics
, the L-function
s of number theory
are expected to have several characteristic properties, one of which is that they satisfy certain functional equation
s. There is an elaborate theory of what these equations should be, much of which is still conjectural. For example, the Riemann zeta function has a functional equation relating its value at the complex number
s with its value at 1 − s. In every case this relates to some value ζ(s) that is only defined by analytic continuation
from the infinite series definition. That is, writingas is conventionalσ for the real part of s, the functional equation relates the cases
and also changes a case with
in the critical strip to another such case, reflected in the line σ = ½. Therefore use of the functional equation is basic, in order to study the zeta-function in the whole complex plane
.
The functional equation in question for the Riemann zeta function takes the simple form
where Z(s) is ζ(s) multiplied by a gamma-factor, involving the gamma function
. This is now read as an 'extra' factor in the Euler product
for the zeta-function, corresponding to the infinite prime. Just the same shape of functional equation holds for the Dedekind zeta function
of a number field K, with an appropriate gamma-factor that depends only on the embeddings of K (in algebraic terms, on the tensor product
of K with the real field
).
There is a similar equation for the Dirichlet L-functions, but this time relating them in pairs:
with χ a primitive Dirichlet character, χ* its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of absolute value
1, of shape
where G(χ) is a Gauss sum
formed from χ. This equation has the same function on both sides if and only if χ is a real character, taking values in {0,1,−1}. Then ε must be 1 or −1, and the case of the value −1 would imply a zero of Λ(s) at s = ½. According to the theory (of Gauss, in effect) of Gauss sums, the value is always 1, so no such simple zero can exist (the function is even about the point).
A unified theory of such functional equations was given by Erich Hecke
, and the theory was taken up again in Tate's thesis
by John Tate
. Hecke found generalised characters of number fields, now called Hecke character
s, for which his proof (based on theta functions) also worked. These characters and their associated L-functions are now understood to be strictly related to complex multiplication
, as the Dirichlet characters are to cyclotomic field
s.
There are also functional equations for the local zeta-function
s, arising at a fundamental level for the (analogue of) Poincaré duality
in étale cohomology
. The Euler products of the Hasse-Weil zeta-function for an algebraic variety
V over a number field K, formed by reducing modulo prime ideal
s to get local zeta-functions, are conjectured to have a global functional equation; but this is currently considered out of reach except in special cases. The definition can be read directly out of étale cohomology theory, again; but in general some assumption coming from automorphic representation theory seems required to get the functional equation. The Taniyama-Shimura conjecture was a particular case of this as general theory. By relating the gamma-factor aspect to Hodge theory
, and detailed studies of the expected ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s of number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
are expected to have several characteristic properties, one of which is that they satisfy certain functional equation
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
s. There is an elaborate theory of what these equations should be, much of which is still conjectural. For example, the Riemann zeta function has a functional equation relating its value at the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s with its value at 1 − s. In every case this relates to some value ζ(s) that is only defined by analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
from the infinite series definition. That is, writingas is conventionalσ for the real part of s, the functional equation relates the cases
- σ > 1 and σ < 0,
and also changes a case with
- 0 < σ < 1
in the critical strip to another such case, reflected in the line σ = ½. Therefore use of the functional equation is basic, in order to study the zeta-function in the whole complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
.
The functional equation in question for the Riemann zeta function takes the simple form
where Z(s) is ζ(s) multiplied by a gamma-factor, involving the gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
. This is now read as an 'extra' factor in the Euler product
Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:...
for the zeta-function, corresponding to the infinite prime. Just the same shape of functional equation holds for the Dedekind zeta function
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK, is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q...
of a number field K, with an appropriate gamma-factor that depends only on the embeddings of K (in algebraic terms, on the tensor product
Tensor product of fields
In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field. On the other hand it is often required to 'join' two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K...
of K with the real field
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
).
There is a similar equation for the Dirichlet L-functions, but this time relating them in pairs:
with χ a primitive Dirichlet character, χ* its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
1, of shape
where G(χ) is a Gauss sum
Gauss sum
In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...
formed from χ. This equation has the same function on both sides if and only if χ is a real character, taking values in {0,1,−1}. Then ε must be 1 or −1, and the case of the value −1 would imply a zero of Λ(s) at s = ½. According to the theory (of Gauss, in effect) of Gauss sums, the value is always 1, so no such simple zero can exist (the function is even about the point).
A unified theory of such functional equations was given by Erich Hecke
Erich Hecke
Erich Hecke was a German mathematician. He obtained his doctorate in Göttingen under the supervision of David Hilbert. Kurt Reidemeister and Heinrich Behnke were among his students....
, and the theory was taken up again in Tate's thesis
Tate's thesis
In number theory, Tate's thesis is the thesis of where he reformulated the work of Erich Hecke on L-series in terms of Fourier analysis on adelic groups....
by John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
. Hecke found generalised characters of number fields, now called Hecke character
Hecke character
In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class ofL-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to...
s, for which his proof (based on theta functions) also worked. These characters and their associated L-functions are now understood to be strictly related to complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
, as the Dirichlet characters are to cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s.
There are also functional equations for the local zeta-function
Local zeta-function
In number theory, a local zeta-functionis a function whose logarithmic derivative is a generating functionfor the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F.-Formulation:...
s, arising at a fundamental level for the (analogue of) Poincaré duality
Poincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
in étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
. The Euler products of the Hasse-Weil zeta-function for an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V over a number field K, formed by reducing modulo prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s to get local zeta-functions, are conjectured to have a global functional equation; but this is currently considered out of reach except in special cases. The definition can be read directly out of étale cohomology theory, again; but in general some assumption coming from automorphic representation theory seems required to get the functional equation. The Taniyama-Shimura conjecture was a particular case of this as general theory. By relating the gamma-factor aspect to Hodge theory
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
, and detailed studies of the expected ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.
See also
- explicit formula (L-function)
- approximate functional equation