Dirichlet beta function

In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.

The Dirichlet beta function

DefinitionEdit

The Dirichlet beta function is defined as

 

or, equivalently,

 

In each case, it is assumed that Re(s) > 0.

Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane:[1]

 

Another equivalent definition, in terms of the Lerch transcendent, is:

 

which is once again valid for all complex values of s.

The Dirichlet beta function can also be written in terms of the Polylogarithm function:

 

Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function

 

but this formula is only valid at positive integer values of  .

Euler product formulaEdit

It is also the simplest example of a series non-directly related to   which can also be factorized as an Euler product, thus leading to the idea of Dirichlet character defining the exact set of Dirichlet series having a factorization over the prime numbers.

At least for Re(s) ≥ 1:

 

where p≡1 mod 4 are the primes of the form 4n+1 (5,13,17,...) and p≡3 mod 4 are the primes of the form 4n+3 (3,7,11,...). This can be written compactly as

 

Functional equationEdit

The functional equation extends the beta function to the left side of the complex plane Re(s) ≤ 0. It is given by

 

where Γ(s) is the gamma function.

Special valuesEdit

Some special values include:

 
 
 

where G represents Catalan's constant, and

 
 
 
 

where   in the above is an example of the polygamma function. More generally, for any positive integer k:

 

where   represent the Euler numbers. For integer k ≥ 0, this extends to:

 

Hence, the function vanishes for all odd negative integral values of the argument.

For every positive integer k:

 [citation needed]

where   is the Euler zigzag number.

Also it was derived by Malmsten in 1842 that

 
s approximate value β(s) OEIS
1/5 0.5737108471859466493572665 A261624
1/4 0.5907230564424947318659591 A261623
1/3 0.6178550888488520660725389 A261622
1/2 0.6676914571896091766586909 A195103
1 0.7853981633974483096156608 A003881
2 0.9159655941772190150546035 A006752
3 0.9689461462593693804836348 A153071
4 0.9889445517411053361084226 A175572
5 0.9961578280770880640063194 A175571
6 0.9986852222184381354416008 A175570
7 0.9995545078905399094963465
8 0.9998499902468296563380671
9 0.9999496841872200898213589
10 0.9999831640261968774055407

There are zeros at -1; -3; -5; -7 etc.

See alsoEdit

ReferencesEdit

  • Glasser, M. L. (1972). "The evaluation of lattice sums. I. Analytic procedures". J. Math. Phys. 14 (3): 409. Bibcode:1973JMP....14..409G. doi:10.1063/1.1666331.
  • J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
  • Weisstein, Eric W. "Dirichlet Beta Function". MathWorld.