Dirichlet's L function

Under Dirichlet L-functions is defined as a family of special mathematical functions , which in analytic number theory , a branch of mathematics , an important role to play. Its namesake, Peter Gustav Lejeune Dirichlet , first used it to prove the so-called Dirichlet prime number theorem . They are usually referred to with the symbol , where is a Dirichlet character and a complex number . ${\displaystyle L(s,\chi )}$ ${\displaystyle \chi }$ ${\displaystyle s}$

For values with a real part greater than 1, all Dirichlet's L functions are defined via a Dirichlet series - namely the Dirichlet generated function of . Is the character also non-principal, i.e. H. it also assumes values ​​other than 0 and 1 in the whole numbers; the series representation even applies to values ​​with a positive real part. By analytic continuation can be a on holomorphic be extended function, wherein in case of a main character in present a pole of the first order. In all other cases a whole continuation is possible. They satisfy important functional equations . ${\displaystyle s\in \mathbb {C} }$ ${\displaystyle \chi (n)}$ ${\displaystyle L(s,\chi )}$ ${\displaystyle \mathbb {C} \setminus \{1\}}$ ${\displaystyle s=1}$ ${\displaystyle L(s,\chi )}$

It is important for number theory that due to the complete multiplicativity of the characters, every Dirichlet L-function can be developed into an Euler product . This provides the crucial information and applications to the theory of prime numbers and gave Dirichlet the means to prove Dirichlet's prime number theorem.

The behavior of Dirichlet's L-functions is widely understood in the areas and . However, their properties are largely unknown within the critical strip and are the subject of significant speculation. Among other things, this concerns the questions of asymptotic growth in the imaginary direction and the zero point distributions that are so important for number theory. According to the current state of knowledge, Dirichlet's L-functions in the strip essentially describe chaos . Areas of application are probability theory and the theory of automorphic forms (especially in the field of the Langlands program ). ${\displaystyle \operatorname {Re} (s)\geq 1}$ ${\displaystyle \operatorname {Re} (s)\leq 0}$ ${\displaystyle 0<\operatorname {Re} (s)<1}$ ${\displaystyle 1/2<\operatorname {Re} (s)<1}$

From the point of view of algebraic number theory , Dirichlet's L-functions are only a special case of a whole class of so-called L-functions . Thus, products of these functions form Dedekind's zeta functions to Abelian extensions . Important special cases of Dirichletscher L functions are the Riemann zeta function and the Dirichlet beta function .

motivation

At the center of number theory , that branch of mathematics that deals with the properties of the natural numbers 1, 2, 3, 4 ..., are the prime numbers 2, 3, 5, 7, 11 .... These are distinguished by the property of having exactly two factors , namely 1 and itself. 1 is not a prime number. Already Euclid could show that there are infinitely many prime numbers, which is why the list 2, 3, 5, 7, 11 ... will never end.

The prime numbers are, so to speak, the atoms of the whole numbers, since every positive whole number can be clearly multiplicatively decomposed into such. For example, 21 = 3 · 7 and 110 = 2 · 5 · 11. Despite this elementary property, after several millennia of mathematics history, no simple pattern is known to which the prime numbers are subject in their sequence. Their nature is one of the most significant open questions in mathematics.

Even if the detailed understanding of the sequence 2, 3, 5, 7, 11 ... is unreachable, certain questions can be answered well with today's methods. This applies, for example, to generalizations of the Euclidean theorem about the infinity of prime numbers. If certain subsets of the set of all prime numbers are infinite, then Euclid's theorem follows. For example, one can ask whether there are infinitely many prime numbers with the property that it is divisible by 4. The first of these prime numbers are 5, 13, 17, 29, 37, 41, ... Dirichlet was able to show, as a consequence of Dirichlet's prime number theorem , that this list never ends. In addition, all lists of the form contains infinitely many prime numbers, as long and except one no common positive divisor have. Accordingly, the list 7, 10007, 20007, 30007, 40007, ... with a = 10000 and b = 7 also contains an infinite number of prime numbers, the first are 7, 10007, 90007, 180007, 240007, 250007, ... ${\displaystyle p}$ ${\displaystyle p-1}$ ${\displaystyle (ak+b)_{k=0,1,2,3,\dotsc }=b,a+b,2a+b,3a+b,\dotsc }$ ${\displaystyle a}$ ${\displaystyle b}$

The Dirichlet L-functions are useful for proving these infinity statements, since the associated mathematical characters can serve to “count” all corresponding prime numbers. This count results in an infinitely large value and that completes the proof.

The prime numbers are not only the subject of basic mathematical research, but also have practical applications. For example, very large prime numbers are used in cryptosystems such as RSA encryption .

Definition and elementary forms of representation

Dirichlet series

A Dirichlet L-function can be defined for each Dirichlet character . This is then given by the Dirichlet series ${\displaystyle \chi }$

${\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.}$

One can show that this series converges in the case of a non-principal character for all complex values with a positive real part . If present, the convergence in the strip is conditional. In each case there is absolute convergence for . ${\displaystyle \chi }$ ${\displaystyle s}$ ${\displaystyle 0<\operatorname {Re} (s)\leq 1}$ ${\displaystyle \operatorname {Re} (s)>1}$

Despite these limitations, the Dirichlet series is used as the basic definition due to its simplicity and its number-theoretical relevance (see Euler product). By means of an analytical continuation (see below) a meaningful calculation for all complex numbers (possibly with ) is possible. ${\displaystyle s}$ ${\displaystyle s\neq 1}$

Euler product

→ Main article : Euler product

Dirichlet characters are completely multiplicative - it applies to all numbers : It follows from this fact that the function in the area of ​​absolute convergence of the Dirichlet series, i.e. exactly for , can be developed into an Euler product: ${\displaystyle \chi }$ ${\displaystyle n,m\in \mathbb {Z} }$ ${\displaystyle \chi (nm)=\chi (n)\chi (m).}$ ${\displaystyle L(s,\chi )}$ ${\displaystyle \operatorname {Re} (s)>1}$

${\displaystyle L(s,\chi )=\prod _{p\,\mathrm {Primzahl} }{\frac {1}{1-\chi (p)p^{-s}}}.}$

This representation is important for number theory. Although some of the L-functions even converge for value , this is in no case true for the Euler product in the entire range , since there is no absolute convergence of the Dirichlet series. For product formation, however, the order of the summands must be interchangeable. ${\displaystyle \mathrm {Re} (s)>0}$ ${\displaystyle \mathrm {Re} (s)\leq 1}$

Primitive characters

The theory of Dirichlet's L-functions for arbitrary characters is reduced to the theory of primitive characters. If a character is induced modulo by a modulo , it follows via the Euler products: ${\displaystyle \chi }$ ${\displaystyle m}$ ${\displaystyle \chi _{1}}$ ${\displaystyle m_{1}}$

${\displaystyle L(s,\chi )=L(s,\chi _{1})\prod _{p|m}(1-\chi _{1}(p)p^{-s}).}$

The back factor is a finite product of Dirichlet series that are very easy to control, which is why important questions such as analytical continuability or zeros only need to be answered for the primitive case.

Analytical continuation

For main characters

For main characters modulo the problem can be solved with ${\displaystyle \chi }$ ${\displaystyle m}$

${\displaystyle L(s,\chi )=\zeta (s)\prod _{p|m}(1-p^{-s})}$

lead back to the Riemann zeta function . It is used that the finite factor behind represents a whole function. The result is that in can be expanded to a completely holomorphic function with a simple pole . ${\displaystyle L(s,\chi )}$ ${\displaystyle \mathbb {C} \setminus \{1\}}$ ${\displaystyle s=1}$

No main characters

If there is no main character, the series converges only for values . Dirichlet's L-functions (to non-principal characters), initially defined only for complex numbers with a positive real part, can be expanded to become completely holomorphic functions. This fact may seem unusual at first, as its Dirichlet series can no longer converge in many places: the expressions do not form a zero sequence for values with a non-positive real part, which is why the necessary criterion for the convergence of a series is violated. In fact, these series are not always available for the definition of Dirichlet's L-functions. ${\displaystyle \chi }$ ${\displaystyle \textstyle \sum {\frac {\chi (n)}{n^{s}}}}$ ${\displaystyle \operatorname {Re} (s)>0}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \textstyle {\frac {\chi (n)}{n^{s}}}}$ ${\displaystyle s}$

An analytic continuation of the area through the series holomorphic function defined is on a larger area holomorphic function on all matches of this. According to the identity theorem for holomorphic functions , such a continuation is always uniquely determined. This means that all values ​​of the function in the extended area are already determined by the Dirichlet series, although here they no longer converge at all points. ${\displaystyle H:=\{s\in \mathbb {C} \mid \operatorname {Re} (s)>0\}}$ ${\displaystyle \textstyle \sum {\frac {\chi (n)}{n^{s}}}}$ ${\displaystyle H\subsetneq D}$ ${\displaystyle H}$ ${\displaystyle L(s,\chi )}$ ${\displaystyle D}$

Functional equation

In the following, denotes the gamma function that generalizes the factorial to complex numbers. Dirichlet's L-functions satisfy all characteristic functional equations that relate the values and . Some authors differentiate between cases (for reasons of clarity) . ${\displaystyle \Gamma (\cdot )}$ ${\displaystyle L(s,\chi )}$ ${\displaystyle L(1-s,{\overline {\chi }})}$ ${\displaystyle \chi (-1)=\pm 1}$

In the following denotes the Gaussian sum of a character. If a primitive character is modulo and , then the identity between meromorphic functions is valid ${\displaystyle {\mathcal {G}}(\chi )}$ ${\displaystyle \chi }$ ${\displaystyle m}$ ${\displaystyle \chi (-1)=1}$

${\displaystyle \pi ^{{\frac {1}{2}}(s-1)}m^{{\frac {1}{2}}(1-s)}\Gamma \left({\frac {1-s}{2}}\right)L(1-s,{\overline {\chi }})={\frac {\sqrt {m}}{{\mathcal {G}}(\chi )}}\pi ^{-{\frac {s}{2}}}m^{\frac {s}{2}}\Gamma \left({\frac {s}{2}}\right)L(s,\chi ).}$

If, on the other hand, is odd, that is , a similar relation follows: ${\displaystyle \chi }$ ${\displaystyle \chi (-1)=-1}$

${\displaystyle \pi ^{{\frac {1}{2}}(s-2)}m^{{\frac {1}{2}}(2-s)}\Gamma \left({\frac {2-s}{2}}\right)L(1-s,{\overline {\chi }})={\frac {\mathrm {i} {\sqrt {m}}}{{\mathcal {G}}(\chi )}}\pi ^{-{\frac {s+1}{2}}}m^{\frac {s+1}{2}}\Gamma \left({\frac {s+1}{2}}\right)L(s,\chi ).}$

Special function values

In the context of certain values ​​of Dirichlet's L-functions, the generalized Bernoulli numbers , which are via ${\displaystyle B_{k,\chi }}$

${\displaystyle \sum _{a=1}^{m}{\frac {\chi (a)z\mathrm {e} ^{az}}{\mathrm {e} ^{mz}-1}}=\sum _{k=0}^{\infty }{\frac {B_{k,\chi }}{k!}}z^{k}}$

let define a major role.

Function values ​​for natural numbers

For the determination of functional values ​​in positive natural places, the key figure

${\displaystyle \delta ={\frac {1-\chi (-1)}{2}}}$

utilized. This is 0 if the character is even, otherwise 1. For positive integers such that is an even number, the following applies to primitive characters modulo : ${\displaystyle k}$ ${\displaystyle k-\delta }$ ${\displaystyle m}$

${\displaystyle L(k,\chi )=(-1)^{1+(k-\delta )/2}{\frac {{\mathcal {G}}(\chi )}{2\mathrm {i} ^{\delta }}}\left({\frac {2\pi }{m}}\right)^{k}{\frac {B_{k,{\overline {\chi }}}}{k!}}.}$

Here called the Gaussian sum of . Is straight, that reduces to ${\displaystyle {\mathcal {G}}(\chi )}$ ${\displaystyle \chi }$ ${\displaystyle \chi }$

${\displaystyle L(2k,\chi )=(-1)^{1+k}{\frac {{\mathcal {G}}(\chi )B_{2k,{\overline {\chi }}}(2\pi )^{2k}}{2m^{2k}(2k)!}}.}$

For odd , however, you have ${\displaystyle \chi }$

${\displaystyle L(2k-1,\chi )=(-1)^{k}{\frac {{\mathcal {G}}(\chi )B_{2k-1,{\overline {\chi }}}(2\pi )^{2k-1}}{2\mathrm {i} m^{2k-1}(2k-1)!}}.}$

In these cases the corresponding L-values ​​can be represented as algebraic multiples of the powers . According to a sentence by Ferdinand von Lindemann, they belong to the transcendent numbers. ${\displaystyle \pi ^{2k}}$

To this day, very little is known about the other values, i.e. for even and odd characters. In these cases , with the exception of the Apery constants , one does not even know whether one of these values is irrational . However, it is believed that they are all irrational. An important special case is and the Catalan's constant ${\displaystyle L(2k-1,\chi )}$ ${\displaystyle L(2k,\chi )}$ ${\displaystyle 2k-1,2k>1}$ ${\displaystyle \zeta (3)}$ ${\displaystyle m=4}$

${\displaystyle G=\beta (2)=1-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-{\frac {1}{11^{2}}}\pm \cdots ,}$

whose irrationality has not yet been proven. But it is known, for example, that an infinite number of values and for are irrational. ${\displaystyle \zeta (2k+1)}$ ${\displaystyle \beta (2k)}$ ${\displaystyle k\in \mathbb {N} }$

For non-positive integers

The values ​​in positive whole places are connected to those in negative whole places via the functional equation. It always applies to primitive characters

${\displaystyle L(1-k,\chi )=-{\frac {B_{k,\chi }}{k}}.}$

From this the trivial zeros of the L-functions can be read off. If is odd, the only exception to this rule is the case , and the trivial character, because it applies . ${\displaystyle k-\delta }$ ${\displaystyle L(1-k,\chi )=0.}$ ${\displaystyle m=1}$ ${\displaystyle k=1}$ ${\displaystyle \zeta (0)=-{\frac {1}{2}}\neq 0}$

The values ${\displaystyle L(1,\chi )}$

There is no main character in this section . Numbers are of particular interest in number theory. They mark the point at which the Euler product “just about” no longer converges. Therefore, handling them tends to be more difficult, especially when it comes to whether . This non-vanishing theorem is a central intermediate step in the proof of Dirichlet's prime number theorem . ${\displaystyle \chi }$ ${\displaystyle L(1,\chi )}$ ${\displaystyle L(1,\chi )\neq 0}$

transcendence

If there is no main character, the value is always a transcendent number. This follows from Baker's theorem and the relation ${\displaystyle \chi }$ ${\displaystyle L(1,\chi )}$

${\displaystyle L(1,\chi )=\sum _{n=1}^{\infty }{\frac {1}{n}}\sum _{k=1}^{m}{\widehat {\chi }}(k)\mathrm {e} ^{2\pi \mathrm {i} kn/m}=-\sum _{k=1}^{m-1}{\widehat {\chi }}(k)\log \left(1-\mathrm {e} ^{2\pi \mathrm {i} k/m}\right).}$

Here are the algebraic numbers and defined by the identity ${\displaystyle {\widehat {\chi }}(k)}$

${\displaystyle \chi (n)=\sum _{k=1}^{m}{\widehat {\chi }}(k)\mathrm {e} ^{2\pi \mathrm {i} nk/m}.}$

The set of seals

If the character is real , then it always applies . Carl Ludwig Siegel was able to tighten this statement by giving an even barrier for the behavior of the varying characters. Is arbitrary, there is a constant such that modulo holds for all real primitive characters ${\displaystyle \chi }$ ${\displaystyle L(1,\chi )>0}$ ${\displaystyle L(1,\chi )}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle C(\varepsilon )>0}$ ${\displaystyle m}$

${\displaystyle L(1,\chi )>C(\varepsilon )m^{-\varepsilon }.}$

literature

• Jörg Brüdern: Introduction to analytical number theory , Springer, 1995.