In analytical number theory , the Dirichlet η function is a special function that is named after the German mathematician Dirichlet (1805-1859). It is related to the Riemannian function .
It is noted with the small Greek letter eta ( ); the Dedekind η function , a modular form , is also called this.
definition
The Dirichlet function is defined for all complexes with a real part greater than 0 using the Dirichlet series :
Although the validity of this expression is limited to complex numbers with a positive real part, it is the starting point for all representations of the function. It can be analytically continued on the whole complex number level, which guarantees a calculation of the function for any arbitrary .
Euler product
The function gets its number-theoretic meaning through its connection to the prime numbers , which are formulary through the Euler product
can express.
Functional equation
The whole identity is:
Connection to the Riemann ζ function
The functional equation between the Dirichletscher and Riemannscher functions can be obtained from the Dirichlet series representations of both functions. The expression is transformed by adding further Dirichlet series to:
The connection follows from this:
which remains fully valid.
Further representations
Integral representation
An integral representation for all contains the gamma function and reads:
-
.
This may come as Mellin transform from being understood. Valid for all is:
Series display
A completely convergent series is obtained with the help of Euler's series transformation :
Product presentation
For everyone , the Hadamard product, named after its discoverer Jacques Hadamard, converges :
It extends over all non-trivial zeros of the function and is simply derived from the Hadamard product of the zeta function .
values
The following applies:
For natural , the Bernoulli numbers apply
For even arguments the general formula applies:
Thus, the numerical value of can always be in the form
write, where and denote two positive integers.
2n
|
p n
|
q n
|
|
2
|
1
|
12
|
0.82246703342411321823 ...
|
4th
|
7th
|
720
|
0.94703282949724591757 ...
|
6th
|
31
|
30240
|
0.98555109129743510409 ...
|
8th
|
127
|
1209600
|
0.99623300185264789922 ...
|
10
|
73
|
6842880
|
0.99903950759827156563 ...
|
12
|
1414477
|
1307674368000
|
0.99975768514385819085 ...
|
14th
|
8191
|
74724249600
|
0.99993917034597971817 ...
|
16
|
16931177
|
1524374691840000
|
0.99998476421490610644 ...
|
18th
|
5749691557
|
5109094217170944000
|
0.99999618786961011347 ...
|
20th
|
91546277357
|
802857662698291200000
|
0.99999904661158152211 ...
|
The first values for odd arguments are
-
(the alternating harmonic series )
zeropoint
From the relation
it is easy to deduce that for all at , as well as additionally at the same places as disappears . This includes both the so-called "trivial" zeros , ie
as well as the "non-trivial" zeros in the strip .
The famous and to this day unproven Riemann Hypothesis states that all non-trivial zeros have the real part 1/2.
Derivation
The derivation of the function is again a Dirichlet series.
A closed expression can be about
and obtained using the product rule .
additional
The relationships between the Dirichlet function and the Riemann function are expressed by the following formula:
or.
The Dirichlet eta function is a special case of the polylogarithm , because:
This makes it a special case of Lerch's zeta function :
Also applies
literature
Individual evidence
-
^ André Voros: More Zeta Functions for the Riemann Zeros (PDF; 182 kB), CEA, Service de Physique Théorique de Saclay (CNRS URA 2306), page 6.
-
↑ Eric W. Weisstein : Dirichlet Lambda Function . In: MathWorld (English).
-
^ J. Spanier, KB Oldham: The Zeta Numbers and Related Functions. In: An Atlas of Functions . Washington, DC: Hemisphere, pp. 25-33, 1987.