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Dirichlet beta function

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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.

Contents

Definition

The Dirichlet beta function is defined as

β ( s ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) s ,

or, equivalently,

β ( s ) = 1 Γ ( s ) ∫ 0 ∞ x s − 1 e − x 1 + e − 2 x d x .

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:

β ( s ) = 4 − s ( ζ ( s , 1 4 ) − ζ ( s , 3 4 ) ) . proof

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

β ( s ) = 2 − s Φ ( − 1 , s , 1 2 ) ,

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

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

β ( s ) = 1 2 s ∑ n = 0 ∞ ( − 1 ) n ( n + 1 2 ) s = 1 ( − 2 ) 2 s ( s − 1 ) ! [ ψ ( s − 1 ) ( 1 4 ) − ψ ( s − 1 ) ( 3 4 ) ] .

Euler product formula

It is also the simplest example of a series non-directly related to ζ ( s ) 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:

β ( s ) = ∏ p ≡ 1   m o d   4 1 1 − p − s ∏ p ≡ 3   m o d   4 1 1 + p − s

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

β ( s ) = ∏ p ≥ 3 p  prime 1 1 − ( − 1 ) p − 1 2 p − s .

Functional equation

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

β ( 1 − s ) = ( π 2 ) − s sin ⁡ ( π 2 s ) Γ ( s ) β ( s )

where Γ(s) is the gamma function.

Special values

Some special values include:

β ( 0 ) = 1 2 , β ( 1 ) = tan − 1 ⁡ ( 1 ) = π 4 , β ( 2 ) = G ,

where G represents Catalan's constant, and

β ( 3 ) = π 3 32 , β ( 4 ) = 1 768 ( ψ 3 ( 1 4 ) − 8 π 4 ) , β ( 5 ) = 5 π 5 1536 , β ( 7 ) = 61 π 7 184320 ,

where ψ 3 ( 1 / 4 ) in the above is an example of the polygamma function. More generally, for any positive integer k:

β ( 2 k + 1 ) = ( − 1 ) k E 2 k π 2 k + 1 4 k + 1 ( 2 k ) ! ,

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

β ( − k ) = E k 2 .

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

For every positive integer k:

β ( 2 k ) = 1 2 ( 2 k − 1 ) ! ∑ m = 0 ∞ ( ( ∑ l = 0 k − 1 ( 2 k − 1 2 l ) ( − 1 ) l A 2 k − 2 l − 1 2 l + 2 m + 1 ) − ( − 1 ) k − 1 2 m + 2 k ) A 2 m ( 2 m ) ! ( π 2 ) 2 m + 2 k ,

where A k is the Euler zigzag number.

Also it was derived by Malmsten in 1842 that

β ′ ( 1 ) = ∑ n = 0 ∞ ( − 1 ) n + 1 ln ⁡ ( 2 n + 1 ) 2 n + 1 = π 4 ( γ − ln ⁡ π ) + π ln ⁡ Γ ( 3 4 )

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

References

Dirichlet beta function Wikipedia
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