Dirichlet eta function

The Dirichletsche function in the complex plane of numbers . ${\displaystyle \eta }$

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 ${\displaystyle \zeta }$ .

It is noted with the small Greek letter eta ( ); the Dedekind η function , a modular form , is also called this. ${\displaystyle \eta }$

definition

The Dirichlet function is defined for all complexes with a real part greater than 0 using the Dirichlet series : ${\displaystyle \eta }$ ${\displaystyle s}$

${\displaystyle \eta (s)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{s}}}=1-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-{\frac {1}{4^{s}}}+{\frac {1}{5^{s}}}-+\cdots .}$

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 . ${\displaystyle \eta }$ ${\displaystyle \eta }$ ${\displaystyle s}$

Euler product

The function gets its number-theoretic meaning through its connection to the prime numbers , which are formulary through the Euler product ${\displaystyle \eta }$ ${\displaystyle \operatorname {Re} s>1}$

${\displaystyle \eta (s)=\left(1-{\frac {1}{2^{s-1}}}\right)\prod _{p\ \mathrm {prim} }{\frac {1}{1-{\frac {1}{p^{s}}}}}=\left(1-{\frac {1}{2^{s-1}}}\right)\cdot {\frac {1}{(1-{\frac {1}{2^{s}}})(1-{\frac {1}{3^{s}}})(1-{\frac {1}{5^{s}}})\cdots }}}$

can express.

Functional equation

The whole identity is: ${\displaystyle \mathbb {C} }$

${\displaystyle \eta (1-s)={\frac {2^{s}-1}{1-2^{s-1}}}\pi ^{-s}\cos \left({\frac {\pi s}{2}}\right)\Gamma (s)\eta (s).}$

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: ${\displaystyle \eta }$ ${\displaystyle \zeta }$ ${\displaystyle \eta }$

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

The connection follows from this:

${\displaystyle \eta (s)=(1-2^{1-s})\cdot \zeta (s),}$

which remains fully valid. ${\displaystyle \mathbb {C} }$

Further representations

Integral representation

An integral representation for all contains the gamma function and reads: ${\displaystyle \operatorname {Re} s>0}$ ${\displaystyle \Gamma (s)}$

${\displaystyle \eta (s)={\frac {1}{\Gamma (s)}}\int \limits _{0}^{\infty }{\frac {x^{s-1}}{e^{x}+1}}{\mathrm {d} x}}$.

This may come as Mellin transform from being understood. Valid for all is: ${\displaystyle {\tfrac {1}{e^{x}+1}}}$ ${\displaystyle s\in \mathbb {C} }$

${\displaystyle \eta (s)={\frac {2^{s-1}-1}{s-1}}-(2^{s}-2)\int \limits _{0}^{\infty }{\frac {\sin(s\arctan x)}{(1+x^{2})^{s/2}(e^{\pi x}+1)}}\mathrm {d} x.}$

Series display

A completely convergent series is obtained with the help of Euler's series transformation : ${\displaystyle \mathbb {C} }$

${\displaystyle \eta (s)=\sum _{n=0}^{\infty }{\frac {1}{2^{n+1}}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}{\frac {1}{(k+1)^{s}}}.}$

Product presentation

For everyone , the Hadamard product, named after its discoverer Jacques Hadamard, converges : ${\displaystyle s\in \mathbb {C} }$

${\displaystyle \eta (s)={\frac {1-2^{1-s}}{2(s-1)\,\Gamma (1+s/2)}}e^{(\ln(2\pi )-1-\gamma /2)s}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)e^{s/\rho }.}$

It extends over all non-trivial zeros of the function and is simply derived from the Hadamard product of the zeta function . ${\displaystyle \rho }$ ${\displaystyle \eta }$

values

The following applies:

${\displaystyle \eta (0)={\tfrac {1}{2}}}$

${\displaystyle \eta (-1)={\tfrac {1}{4}}.}$

For natural , the Bernoulli numbers apply ${\displaystyle k}$ ${\displaystyle B_{k}}$

${\displaystyle \eta (1-k)={\frac {2^{k}-1}{k}}B_{k}.}$

For even arguments the general formula applies: ${\displaystyle 2n=2,4,6,8,\dots }$

${\displaystyle \eta (2n)=(-1)^{n+1}{\frac {2^{2n-1}-1}{(2n)!}}B_{2n}\pi ^{2n}.}$

Thus, the numerical value of can always be in the form ${\displaystyle \eta (2n)}$

${\displaystyle \eta (2n)={\frac {p_{n}}{q_{n}}}\pi ^{2n}}$

write, where and denote two positive integers. ${\displaystyle p_{n}}$ ${\displaystyle q_{n}}$

2n	p n	q n	${\displaystyle \eta (2n)}$
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

${\displaystyle \eta (1)=\ln 2}$(the alternating harmonic series )

${\displaystyle \eta (3)={\frac {3}{4}}\zeta (3)}$

${\displaystyle \eta (5)={\frac {15}{16}}\zeta (5).}$

zeropoint

From the relation

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

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 ${\displaystyle \eta (s)}$ ${\displaystyle m\in \mathbb {Z} \setminus \{0\}}$ ${\displaystyle s_{m}=1+{\tfrac {2\pi mi}{\ln 2}}}$ ${\displaystyle \zeta (s)}$ ${\displaystyle s=-2,-4,-6,-8,\dots }$

${\displaystyle \eta (-2)=\eta (-4)=\eta (-6)=\eta (-8)=\cdots =0,}$

as well as the "non-trivial" zeros in the strip . ${\displaystyle \{s\in \mathbb {C} |0<\operatorname {Re} s<1\}}$

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. ${\displaystyle \eta }$ ${\displaystyle \operatorname {Re} s>0}$

${\displaystyle \eta '(s)=\sum _{n=2}^{\infty }(-1)^{n}{\frac {\ln n}{n^{s}}}.}$

A closed expression can be about

${\displaystyle \eta '(s)={\frac {\mathrm {d} }{\mathrm {d} s}}(1-2^{1-s})\zeta (s)={\frac {}{}}2^{1-s}(\ln 2)\zeta (s)+(1-2^{1-s})\zeta ^{\prime }(s),}$

and obtained using the product rule .

additional

The relationships between the Dirichlet function and the Riemann function are expressed by the following formula: ${\displaystyle \eta }$ ${\displaystyle \lambda }$ ${\displaystyle \zeta }$

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

or.

${\displaystyle \zeta (v)+\eta (v)=2\lambda (v).}$

The Dirichlet eta function is a special case of the polylogarithm , because:

${\displaystyle \eta (x)=-\mathrm {Li} _{x}(-1).}$

This makes it a special case of Lerch's zeta function :

${\displaystyle \eta (s)=\Phi (-1,s,1).}$

Also applies

${\displaystyle \int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {[-\ln(xy)]^{s}}{1+xy}}\;\mathrm {d} x\,\mathrm {d} y=\Gamma (s+2)\eta (s+2).}$

literature

• Eric W. Weisstein : Dirichlet Eta Function . In: MathWorld (English).
• Milton Abramowitz , Irene Stegun : Handbook of Mathematical Functions , New York: Dover, 1972.
• Konrad Knopp [1922]: Theory and Application of the Infinite Series . Dover, 1990, ISBN 0-486-66165-2 . .

Individual evidence

1. ^ André Voros: More Zeta Functions for the Riemann Zeros (PDF; 182 kB), CEA, Service de Physique Théorique de Saclay (CNRS URA 2306), page 6.
2. ↑ Eric W. Weisstein : Dirichlet Lambda Function . In: MathWorld (English).
3. ^ J. Spanier, KB Oldham: The Zeta Numbers and Related Functions. In: An Atlas of Functions . Washington, DC: Hemisphere, pp. 25-33, 1987.