Dirichlet beta function
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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.
Definition
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 formula
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 equation
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. It was conjectured by Euler in 1749 and proved by Malmsten in 1842 (see Blagouchine, 2014).
Specific values
For every positive odd integer , the following equation holds:[2]
where is the n-th Euler Number. This yields:
For negative odd integers, the function is zero:
For every negative even integer it holds[2]:
- .
It further is:
- .
About the values of the Dirichlet beta function at positive even integers not much is known (similarly to the Riemann zeta function at odd integers greater than 3). The number is known as Catalan's constant.
It has been proven that infinitely many numbers of the form and at least one of the numbers are irrational.[3][4]
The number may be given in terms of the polygamma function:
For every positive integer k:
- [citation needed]
where is the Euler zigzag number.
Also it was derived by Malmsten in 1842 (see Blagouchine, 2014) 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 | A258814 |
8 | 0.9998499902468296563380671 | A258815 |
9 | 0.9999496841872200898213589 | A258816 |
10 | 0.9999831640261968774055407 |
See also
References
- ^ Dirichlet Beta – Hurwitz zeta relation, Engineering Mathematics
- ^ a b Weisstein, Eric W. "Dirichlet Beta Function". mathworld.wolfram.com. Retrieved 2024-08-08.
- ^ Zudilin, Wadim (2019-05-31). "Arithmetic of Catalan's constant and its relatives". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 89 (1): 45–53. doi:10.1007/s12188-019-00203-w. ISSN 0025-5858.
- ^ Rivoal, T.; Zudilin, W. (2003-08-01). "Diophantine properties of numbers related to Catalan's constant". Mathematische Annalen. 326 (4): 705–721. doi:10.1007/s00208-003-0420-2. ISSN 1432-1807.
- Blagouchine, I. V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". Ramanujan J. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5.
- 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.