The 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

${\displaystyle \beta (s)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{s}}},}$

or, equivalently,

${\displaystyle \beta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}e^{-x}}{1+e^{-2x}}}\,dx.}$

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:^[1]

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

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

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

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:

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

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

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

but this formula is only valid at positive integer values of ${\displaystyle s}$.

Euler product formula

It is also the simplest example of a series non-directly related to ${\displaystyle \zeta (s)}$ 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:

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

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

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

Functional equation

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

${\displaystyle \beta (1-s)=\left({\frac {\pi }{2}}\right)^{-s}\sin \left({\frac {\pi }{2}}s\right)\Gamma (s)\beta (s)}$

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:

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

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

${\displaystyle \beta (2)\;=\;G,}$

where G represents Catalan's constant, and

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

${\displaystyle \beta (4)\;=\;{\frac {1}{768}}\left(\psi _{3}\left({\frac {1}{4}}\right)-8\pi ^{4}\right),}$

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

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

where ${\displaystyle \psi _{3}(1/4)}$ 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:

${\displaystyle \beta (2k)={\frac {1}{2(2k-1)!}}\sum _{m=0}^{\infty }\left(\left(\sum _{l=0}^{k-1}{\binom {2k-1}{2l}}{\frac {(-1)^{l}A_{2k-2l-1}}{2l+2m+1}}\right)-{\frac {(-1)^{k-1}}{2m+2k}}\right){\frac {A_{2m}}{(2m)!}}{\left({\frac {\pi }{2}}\right)}^{2m+2k},}$

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

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

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

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

See also

• Hurwitz zeta function
• Dirichlet eta function
• Polylogarithm

References

• 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.