Dirichlet series

Dirichlet series , named after Peter Gustav Lejeune Dirichlet , are series that are used in analytical number theory to investigate number theoretic functions with methods from analysis , especially function theory. Many open number theoretic questions have become accessible to an "approximate solution" (through estimates) through this connection, such as questions about the distribution of prime numbers .

Convergent Dirichlet series are also interesting as analytical functions , detached from number theoretic problems, as an object of investigation, as they are closely related to power series and allow a similarly “natural” representation of analytical functions.

Definition and formal characteristics

A Dirichlet series is a series of form

${\displaystyle F(s)=\sum _{n=1}^{\infty }{\frac {f(n)}{n^{s}}},}$

With ${\displaystyle s=\sigma +it\in \mathbb {C} .}$

This series converges absolutely for certain coefficient sequences and complex numbers . The product of two such absolutely convergent Dirichlet series is again an absolutely convergent Dirichlet series, the coefficients result from the convolution of the coefficient sequences as number-theoretic functions . The multiplication of absolutely convergent Dirichlet series corresponds to the convolution of their coefficients. ${\displaystyle f(n)}$ ${\displaystyle s}$

Occasionally one can find the more general definition in the literature ( e.g. in Zagier )

${\displaystyle F(s)=\sum _{n=1}^{\infty }f(n)e^{-\lambda _{n}s},}$ With ${\displaystyle \lambda _{1}\leq \lambda _{2}\leq \lambda _{3}...\rightarrow \infty .}$

With this you get the first definition again, with you get ${\displaystyle \lambda _{n}=\operatorname {log} n}$ ${\displaystyle \lambda _{n}=n}$

${\displaystyle F(s)=\sum _{n=1}^{\infty }f(n)e^{-ns}=\sum _{n=1}^{\infty }f(n)z^{n}}$with , ${\displaystyle z=e^{-s}}$

thus an ordinary power series .

The space of the formal Dirichlet series is provided with a multiplication by transferring the multiplication rule valid for absolutely convergent series to any (including non-convergent) Dirichlet series (for this construction, compare the analogous concept formation formal power series ).

This makes the space of the formal Dirichlet series with pointwise addition, scalar multiplication and convolution isomorphic (as a ring and algebra ) to the number-theoretic functions and inherits all structural properties of this space.

Isomorphism assigns the formal Dirichlet series to every number-theoretic function , whose coefficient sequence it is. This Dirichlet series is then called the Dirichlet series generated by . ${\displaystyle f(n)}$ ${\displaystyle F(s)}$ ${\displaystyle f(n)}$

Identity set

If two ordinary Dirichlet series and , both of which converge on a half-plane , agree on an open subset , then it already follows that they are completely identical and that all of their coefficients agree exactly. Then we have so and for all . ${\displaystyle F(s)=\sum a(n)/n^{s}}$ ${\displaystyle G(s)=\sum b(n)/n^{s}}$ ${\displaystyle H}$ ${\displaystyle U\subset H}$ ${\displaystyle H}$ ${\displaystyle F=G}$ ${\displaystyle a(n)=b(n)}$ ${\displaystyle n\in \mathbb {N} }$

Convergent Dirichlet series

For every Dirichlet series that converges somewhere, but not everywhere, there is a real number , so that the series converges in the half plane ( is the real part of ) and diverges in the half plane . No general statement can be made about the behavior on the straight . If the Dirichlet series converges everywhere or nowhere, or is set and in all cases the convergence is called the abscissa of the Dirichlet series. ${\displaystyle \sigma _{0}}$ ${\displaystyle \operatorname {Re} (s)>\sigma _{0}}$ ${\displaystyle \operatorname {Re} (s)}$ ${\displaystyle s}$ ${\displaystyle \operatorname {Re} (s)<\sigma _{0}}$ ${\displaystyle \operatorname {Re} (s)=\sigma _{0}}$ ${\displaystyle \sigma _{0}=-\infty }$ ${\displaystyle \sigma _{0}=\infty }$ ${\displaystyle \sigma _{0}\in [-\infty ,\infty ]}$

Similar to how the radius of convergence can be calculated in the case of power series , in the case of Dirichlet series the convergence abscissa with a limit superior can also be determined from their coefficient sequence:

Is divergent, so is ${\displaystyle \sum _{n=1}^{\infty }f(n)}$

${\displaystyle \sigma _{0}=\limsup _{N\rightarrow \infty }{\frac {\log(|f(1)+\ldots +f(N)|)}{\log N}}}$.

If, however, is convergent, then is ${\displaystyle \sum _{n=1}^{\infty }f(n)}$

${\displaystyle \sigma _{0}=\limsup _{N\rightarrow \infty }{\frac {\log {(|\sum _{n=N}^{\infty }f(n)|)}}{\log {N}}}}$.

Analytical properties

In its half-plane of convergence , the Dirichlet series is compactly convergent and represents a holomorphic function there . ${\displaystyle \operatorname {Re} (s)>\sigma _{0}}$ ${\displaystyle F(s)}$

The derivatives of the holomorphic function determined in this way can be obtained by differentiation by term. Its -th derivative is the Dirichlet series ${\displaystyle F}$ ${\displaystyle k}$

${\displaystyle F^{(k)}(s)=(-1)^{k}\sum _{n=1}^{\infty }{\frac {f(n)\cdot (\log n)^{k}}{n^{s}}}}$.

Euler products

→ Main article : Euler product

Dirichlet series with multiplicative number theoretic functions as coefficients can be represented as Euler products. If it is a multiplicative number theoretic function and the Dirichlet series F ( s ) it generates converges absolutely for the complex number s , then the following applies ${\displaystyle f(n)}$

${\displaystyle F(s)=\prod _{p\ {\rm {prim}}}\sum _{k=0}^{\infty }{\frac {f(p^{k})}{p^{ks}}}}$.

In the case of a fully multiplicative function, this product is simplified to

${\displaystyle F(s)=\prod _{p\ \operatorname {prim} }{\frac {1}{1-f(p)p^{-s}}}}$.

These infinite products over all prime numbers are called Euler products. The value of these products is defined as the limit of the sequence of finite products , which is created by this product on prime numbers below a barrier N extends. ${\displaystyle \lim _{N\to \infty }P_{N}}$ ${\displaystyle P_{N}}$

Important Dirichlet series

Riemann ζ function

→ Main article : Riemann ζ function

The most famous Dirichlet series is the Riemann ζ function:

${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=1+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+\cdots }$.

It is generated by the number-theoretic 1-function (with for all ). Since this function is completely multiplicative, the zeta function has the Euler product representation ${\displaystyle I^{0}}$ ${\displaystyle I^{0}(n)=1}$ ${\displaystyle n}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p\ {\rm {prim}}}{\frac {1}{1-p^{-s}}}.}$

Dirichlet series of the divider function

The divisor function (also more precisely the number of divisors function ) , which assigns the number of its positive factors to a natural number , is the "convolution square" of the 1 function. ${\displaystyle d(n)}$ ${\displaystyle n}$

${\displaystyle \!\ d(n)=\sum _{d|n}1=(I^{0}*I^{0})(n)}$,

the Dirichlet series assigned to it is therefore the square of the zeta function:

${\displaystyle \sum _{n=1}^{\infty }{\frac {d(n)}{n^{s}}}=\zeta ^{2}(s)}$.

Dirichlet series of the Möbius function

The Möbius function is multiplicative with for . So the Dirichlet series generated by it has the Euler product ${\displaystyle \mu (n)}$ ${\displaystyle \mu (p^{k})=0}$ ${\displaystyle k>1}$ ${\displaystyle M(s)}$

${\displaystyle M(s)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=\prod _{p\ {\rm {prim}}}(1-p^{-s})={\frac {1}{\zeta (s)}}}$.

The relation is transferred to the associated number theoretic functions and means there: ${\displaystyle M(s)\cdot \zeta (s)=1}$

${\displaystyle \mu *I^{0}(n)=\sum _{d|n}\mu (d)={\begin{cases}1&\mathrm {falls} \ n=1\\0&\mathrm {sonst} \end{cases}}}$.

Dirichletsche L-rows

L series is a forwarding to this article. For other meanings see L series .

The L series also introduced by Dirichlet

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

are generated by a Dirichlet character . These series play an important role in the proof of Dirichlet's theorem about the existence of an infinite number of prime numbers in arithmetic progressions . Since Dirichlet characters are completely multiplicative, the L-series can be represented as Euler products ${\displaystyle \chi }$

${\displaystyle L(s,\chi )=\prod _{p\ {\rm {prim}}}{\frac {1}{1-\chi (p)p^{-s}}}}$

and for the main character modulo k applies: ${\displaystyle \chi =\chi _{1}}$

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

The L-series generalize the Riemann zeta function. → About the zeros of L-series there is the generalized Riemann Hypothesis, which has not yet been proven .

Hecke gave a generalization with size characters instead of Dirichlet characters (also called Hecke L-series, but see below for a further definition).

Dirichlet series of the Mangoldt function

The von Mangoldt function plays a role in proving the prime number theorem . This number theoretic function is defined as ${\displaystyle \Lambda (n)}$

${\displaystyle \Lambda (n)={\begin{cases}\log {p}&\mathrm {falls} \ n=p^{m},\ p\ \mathrm {prim} ,\ m\in \mathbb {N} \\0&\mathrm {sonst,} \end{cases}}}$,

the Dirichlet series it generates can be expressed using the zeta function:

${\displaystyle -\sum _{n=1}^{\infty }{\frac {\Lambda (n)}{n^{s}}}={\frac {\zeta '(s)}{\zeta (s)}}}$.

Dirichlet's lambda function

The Dirichlet lambda function is the L series that goes through

${\displaystyle \lambda (s)=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{s}}};\;s\neq 1}$ is defined.

It can be represented by the Riemann zeta function as

${\displaystyle \lambda (s)=(1-2^{-s})\cdot \zeta (s).}$

It can be calculated in closed form at the points where this is possible for the zeta function, i.e. for even positive numbers.The following relationship exists with the Dirichlet eta function : ${\displaystyle s\in 2\mathbb {N} .}$

${\displaystyle {\frac {\zeta (s)}{2^{s}}}={\frac {\lambda (s)}{2^{s}-1}}={\frac {\eta (s)}{2^{s}-2}}.}$

Dirichlet series of Euler's φ-function

The Euler φ function is multiplicative with

${\displaystyle \varphi (p^{k})=p^{k}-p^{k-1}}$for . ${\displaystyle k\geq 1}$

The Euler product of the Dirichlet series generated by it is

${\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)}{n^{s}}}=\prod _{p\ {\rm {prim}}}{\frac {1-p^{-s}}{1-p^{1-s}}}={\frac {\zeta (s-1)}{\zeta (s)}}}$.

Dirichlet series of the generalized partial sum function

The generalized divisional sum function is multiplicative and for prime powers is ${\displaystyle \sigma _{k}(n)}$

${\displaystyle \sigma _{k}(p^{m})={\frac {1-p^{k(m+1)}}{1-p^{k}}}}$.

Hence the Dirichlet series of has the Euler product representation: ${\displaystyle \sigma _{k}}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{k}(n)}{n^{s}}}=\prod _{p\ {\rm {prim}}}{\frac {1}{(1-p^{-s})(1-p^{k-s})}}=\zeta (s)\zeta (s-k).}$

Dirichlet series and modular forms

Erich Hecke found a connection (Hecke correspondence) between Dirichlet series, which satisfy certain Euler product and functional equations, and modular forms , see Hecke operator . The Hecke L-rows defined by him are formed with the Fourier coefficients of the modular shapes. However, these are to be distinguished from the Dirichlet series, which are similar to Dirichlet L series and are also called Hecke L series, with size characters according to Hecke.

folding

The convolution of two number theoretic functions induces a formal ring homomorphism from the ring of number theoretic functions into the ring of formal Dirichlet series via ${\displaystyle (f*g)(n):=\sum _{d|n}f(d)g(n/d)}$

${\displaystyle D_{f*g}=D_{f}\cdot D_{g},}$

wherein the at corresponding Dirichlet rows designate. ${\displaystyle D_{f},D_{g}}$ ${\displaystyle f,g}$

example

One finds, for example, the relation:

${\displaystyle \zeta (s)^{2}=\sum _{n=1}^{\infty }{\frac {d(n)}{n^{s}}}=1+{\frac {2}{2^{s}}}+{\frac {2}{3^{s}}}+{\frac {3}{4^{s}}}+{\frac {2}{5^{s}}}+{\frac {4}{6^{s}}}+{\frac {2}{7^{s}}}+\ldots ,}$

where is the number of divisors function that counts how many natural factors a number has. This result is obtained by systematically multiplying the square of the Dirichlet series of the zeta function. Since it is the product of two (convergent) Dirichlet series, it can, as described above, again be represented by a Dirichlet series. ${\displaystyle d(n)}$ ${\displaystyle n}$

${\displaystyle \zeta (s)^{2}=\left(\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\right)^{2}=\left(\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\right)\cdot \left(\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}\right)=\sum _{n=1}^{\infty }\sum _{k=1}^{\infty }{\frac {1}{n^{s}}}\cdot {\frac {1}{k^{s}}}=\sum _{n=1}^{\infty }\sum _{k=1}^{\infty }{\frac {1}{(n\cdot k)^{s}}}=\sum _{m=1}^{\infty }{\frac {d(m)}{m^{s}}}}$

The Dirichlet series generated from this convolution now has a new number theoretic function, which is referred to as. The sum index is chosen as to avoid confusion. The penultimate step of the evaluation now shows that the value of can be obtained from the number of all natural number pairs for which applies. This reduces the question of the value of to how many factors the number in question has. ${\displaystyle d(m)}$ ${\displaystyle m}$ ${\displaystyle d(m)}$ ${\displaystyle (n,k)}$ ${\displaystyle n\cdot k=m}$ ${\displaystyle d(m)}$ ${\displaystyle m}$

See also

• Mellin transformation

Web links

• Dirichlet Series on PlanetMath
• Eric W. Weisstein : Dirichlet series . In: MathWorld (English).
• Eric W. Weisstein : Dirichlet L-Series . In: MathWorld (English).

literature

• Tom Mike Apostol : Modular Functions and Dirichlet Series in Number Theory . Springer Verlag New York u. a. 1990, ISBN 0-387-97127-0
• Jörg Brüdern : Introduction to analytical number theory . Springer Verlag, Berlin a. a. 1995, ISBN 3-540-58821-3
• Graham James Oscar Jameson: The Prime Number Theorem . Cambridge University Press, Cambridge u. a. 2004, ISBN 0-521-89110-8
• Konrad Knopp : Theory and Application of Infinite Series . 2nd Edition. In: The basic teachings of the mathematical sciences in individual presentations with special consideration of the areas of application . Springer Verlag, Berlin a. a. 1996, ISBN 3-540-59111-7
• Max Koecher , Aloys Krieg : Elliptical functions and modular forms , 2nd edition, Springer, Berlin (2007) ISBN 978-3-540-49324-2
• Don Zagier : Zeta functions and square solids . Springer Verlag Berlin u. a. 1981, ISBN 3-540-10603-0