Series L of Dirichlet

In Mathematical, a series L of Dirichlet , is a series of the plan Plan complex used in analytical Théorie of the numbers.

By analytical Prolongation, this function can be wide with a Fonction méromorphe on the plan complexes whole.

It is built starting from a Caractère of Dirichlet and, if the character is commonplace, the function L of Dirichlet is identified with the Fonction zeta of Riemann.

These properties enable him to show the theorem on the prime numbers in the arithmetic progressions.

It named thus in the honor of the German Mathematician Johann Peter Gustav Lejeune Dirichlet (1805 - 1859) .

Definitions

That is to say χ a character of Dirichlet modulo Q where Q is a strictly positive entirety and S a complex number of real part higher than 1 .

• the series L of Dirichlet for the character χ at the point S , noted L ( S , χ), is given by the following formula:

$L\; (S,\; \backslash \; chi)\; =\; \backslash \; sum\_\; \{n=1\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{\backslash \; chi\; (N)\}\{n^s\}$

• the analytical prolongation of the series L of Dirichlet for the character χ is called Fonction L of Dirichlet and is still noted L ( S , χ).

Behavior at the point one

The behavior of the series at the point a is the key of the Théorème of the arithmetic progression. This is why definite Dirichlet these series. Here, NR indicates the driver of the studied characters and χ₀ the principal character.

* the point a is a pole of any principal nature.

* Any character nonprincipal is definite and analytical on the half Plan complex of strictly positive real part.

What means that she does not admit a pole on this area.

* the point a is not root of any series of Dirichlet.

Zeros of the functions L of Dirichlet

If $\backslash \; chi\; \backslash ,$ is a primitive character with $\backslash \; chi\; (-\; 1)\; =\; 1\; \backslash ,$, then only the zeros of $L\; (S,\; \backslash \; chi)\; \backslash ,$ with Re ( S ) <0 are the negative even entireties. If $\backslash \; chi\; \backslash ,$ is a primitive character with $\backslash \; chi\; (-\; 1)\; =\; -1\; \backslash ,$, then only the zeros of $L\; (S,\; \backslash \; chi)\; \backslash ,$ with Re ( S ) <0 are the negative odd entireties.

Until the possible existence of a Zero of Siegel, the areas without zero including and beyond the line Re ( S ) =1 similar to the function zeta of Riemann are of existence known for all the functions L of Dirichlet.

In the same way that the function of zeta of Riemann is conjectured like obeying the Hypothèse of Riemann, the functions L of Dirichlet are conjectured like obeying the generalized Hypothèse of Riemann.

Functional equation

Let us suppose that $\backslash \; chi\; \backslash ,$ is a primitive character of module K . Defining

$\backslash \; varepsilon\; (S,\; \backslash \; chi)\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; pi\}\; \{K\}\; \backslash \; right)\; ^\; \{-\; (s+a)/2\}$

\ Gamma \ left (\ frac {s+a} {2} \ right) L (S, \ chi),

where $\backslash \; Gamma\; \backslash ,$ indicates the Fonction gamma and the symbol has is given by

one with the functional equation

$\backslash \; varepsilon\; (1-s,\; \backslash \; overline\; \{\backslash \; chi\})\; =\; \backslash \; frac\; \{i^ak^\; \{1/2\}\}\; \{\backslash \; tau\; (\backslash \; chi)\}\backslash \; varepsilon\; (S,\; \backslash \; chi).$

Here, we wrote $\backslash \; tau\; (\backslash \; chi)\; \backslash ,$ for the Somme of Gauss

$\backslash \; sum\_\; \{n=1\}\; ^k\; \backslash \; chi\; (N)\; \backslash \; exp\; (2\; \backslash \; pi\; im/q)\; \backslash ,$.

Note: $|\backslash \; tau\; (\backslash \; chi)|\; =\; k^\; \{\backslash \; frac\; \{1\}\; \{2\}\}\; \backslash ,$.

Relationship to the function zeta of Hurwitz

The functions L of Dirichlet can be written like a linear combination of functions zeta of Hurwitz to rational values. By fixing an entirety $k\; \backslash \; Ge\; 1\; \backslash ,$, the functions L of Dirichlet for the characters modulo K are linear combinations, with constant coefficients, of $\backslash \; zeta\; (S,\; Q)\; \backslash ,$ where Q = m / K and m = 1,2,…, K . This means that the function zeta of Hurwitz for rational a Q has analytical properties which are closely related to the functions L of Dirichlet. Precisely, that is to say $\backslash \; chi\; \backslash ,$ a character modulo K . Then, we can write his function L of Dirichlet in the form

$L\; (\backslash \; chi,\; S)\; =\; \backslash \; sum\_\; \{n=1\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{\backslash \; chi\; (N)\}\{n^s\}$

\ frac {1} {k^s} \ sum_ {m1} ^k \ chi (m) \; \ zeta \ left (S, \ frac {m} {K} \ right) .

In particular, the function L of Dirichlet of the character modulo 1 gives us the function zeta of Riemann:

$\backslash \; zeta\; (S)\; =\; \backslash \; frac\; \{1\}\; \{k^s\}\; \backslash \; sum\_\; \{m=1\}\; ^k\; \backslash \; zeta\; \backslash \; left\; (S,\; \backslash \; frac\; \{m\}\; \{K\}\; \backslash \; right)$.

External bonds and references

External bonds

• Johann Peter Gustav Lejeune Dirichlet by the University of St Andrew
• The life and work off Dirichlet by Jürgen Elstrodt
• Infinitely many premiums, with analysis by the University of Montreal de Andrew Granville and K. Soundararajan
• Dirichlet' S theorem one premiums in arithmetic progression by IMo

References

• Jean-Benoit Bost, Pierre Colmez and Philippe Biane the function Zeta , Editions of the Paris Polytechnic school 2002 ISBN 2730210113
• Harold Davenport' S Multiplicative number theory , 3^ème edt Springer 2000 ISBN 0387950974
• Karatsuba BASIC analytic number theory , Springer-Verlag 1993 ISBN 0-387-53345-1
• S.J. Patterson Year Introduction to the Theory off the Riemann Zeta-Function Cambridge University Close 1995 ISBN 0521499054.

Random links:

Sütterlin | Viewer of images | Viéthorey | Dollar (symbol) | Lou Boudreau | transfert_de_données_de_Noeud-à-noeud

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org