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.

Definition
The Dirichlet beta function is defined as


 * $$\beta(s) = \sum_{n=0}^\infty \frac{(-1)^n} {(2n+1)^s},$$

or, equivalently,


 * $$\beta(s) = \frac{1}{\Gamma(s)}\int_0^{\infty}\frac{x^{s-1}e^{-x}}{1 + e^{-2x}}\,dx.$$

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:


 * $$\beta(s) = 4^{-s} \left( \zeta\left(s,{1 \over 4}\right)-\zeta\left( s, {3 \over 4}\right) \right).$$

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


 * $$\beta(s) = 2^{-s} \Phi\left(-1,s,{{1} \over {2}}\right),$$

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:


 * $$\beta(s) = \frac{i}{2} \left(\text{Li}_s(-i)-\text{Li}_s(i)\right).$$

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


 * $$\beta(s) =\frac{1}{2^s} \sum_{n=0}^\infty\frac{(-1)^{n}}{\left(n+\frac{1}{2}\right)^{s}}=\frac1{(-4)^s(s-1)!}\left[\psi^{(s-1)}\left(\frac{1}{4}\right)-\psi^{(s-1)}\left(\frac{3}{4}\right)\right]$$

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

Euler product formula
It is also the simplest example of a series non-directly related to $$\zeta(s)$$ 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:


 * $$ \beta(s) = \prod_{p \equiv 1 \ \mathrm{mod} \ 4} \frac{1}{1 - p^{-s}} \prod_{p \equiv 3 \ \mathrm{mod} \ 4} \frac{1}{1 + p^{-s}} $$

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


 * $$\beta(s) = \prod_{p>2\atop p \text{ prime}} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}}.$$

Functional equation
The functional equation extends the beta function to the left side of the complex plane Re(s) ≤ 0. It is given by
 * $$\beta(1-s)=\left(\frac{\pi}{2}\right)^{-s}\sin\left(\frac{\pi}{2}s\right)\Gamma(s)\beta(s)$$

where Γ(s) is the gamma function. It was conjectured by Euler in 1749 and proved by Malmsten in 1842 (see Blagouchine, 2014).

Special values
Some special values include:


 * $$\beta(0)= \frac{1}{2}, $$


 * $$\beta(1)\;=\;\arctan(1)\;=\;\frac{\pi}{4}, $$


 * $$\beta(2)\;=\;G,$$

where G represents Catalan's constant, and


 * $$\beta(3)\;=\;\frac{\pi^3}{32},$$


 * $$\beta(4)\;=\;\frac{1}{768}\left(\psi_3\!\left(\frac{1}{4}\right)-8\pi^4\right),$$


 * $$\beta(5)\;=\;\frac{5\pi^5}{1536},$$


 * $$\beta(7)\;=\;\frac{61\pi^7}{184320},$$

where $$\psi_3(1/4)$$ in the above is an example of the polygamma function.



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

For every positive integer k:
 * $$\beta(2k)=\frac{1}{2(2k-1)!}\sum_{m=0}^\infty\left(\left(\sum_{l=0}^{k-1}\binom{2k-1}{2l}\frac{(-1)^{l}A_{2k-2l-1}}{2l+2m+1}\right)-\frac{(-1)^{k-1}}{2m+2k}\right)\frac{A_{2m}}{(2m)!}{\left(\frac{\pi}{2}\right)}^{2m+2k},$$

where $$A_{k}$$ is the Euler zigzag number.

Also it was derived by Malmsten in 1842 (see Blagouchine, 2014) that



\beta'(1)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\ln(2n+1)}{2n+1} \,=\,\frac{\pi}{4}\big(\gamma-\ln\pi) +\pi\ln\Gamma\left(\frac{3}{4}\right) $$

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