Dirichlet beta function

The Dirichlet beta function is defined as

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

or, equivalently,

:$\backslash beta(s)\; =\; \backslash frac\{1\}\{\backslash Gamma(s)\}\backslash int\_0^\{\backslash infty\}\backslash frac\{x^\{s-1\}e^\{-x\}\}\{1\; +\; e^\{-2x\}\}\backslash ,dx.$

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:[http://engineeringandmathematics.blogspot.co.nz/2012/09/dirichlet-beta-hurwitz-zeta-relation.html Dirichlet Beta – Hurwitz zeta relation], Engineering Mathematics

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

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

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

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:

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

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

:$\backslash beta(s)\; =\backslash frac\{1\}\{2^s\}\; \backslash sum\_\{n=0\}^\backslash infty\backslash frac\{(-1)^\{n\}\}\{\backslash left(n+\backslash frac\{1\}\{2\}\backslash right)^\{s\}\}=\backslash frac1\{(-4)^s(s-1)!\}\backslash left[\backslash psi^\{(s-1)\}\backslash left(\backslash frac\{1\}\{4\}\backslash right)-\backslash psi^\{(s-1)\}\backslash left(\backslash frac\{3\}\{4\}\backslash right)\backslash right]$

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

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

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

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

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

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

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

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

For every odd positive integer $2n+1$, the following equation holds:{{Cite web |last=Weisstein |first=Eric W. |title=Dirichlet Beta Function |url=https://mathworld.wolfram.com/ |access-date=2024-08-08 |website=mathworld.wolfram.com |language=en}}

:$\backslash beta(2n+1)\backslash ;=\backslash ;\backslash frac\{(-1)^n\; E\_\{2n\}\}\{2(2n)!\}\backslash left(\backslash frac\backslash pi2\backslash right)^\{2n+1\}$

where $E\_n$ is the n-th Euler Number. This yields:

:$\backslash beta(1)\backslash ;=\backslash ;\backslash frac\{\backslash pi\}\{4\},$

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

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

:$\backslash beta(7)\backslash ;=\backslash ;\backslash frac\{61\backslash pi^7\}\{184320\}$

For the values of the Dirichlet beta function at even positive integers no elementary closed form is known, and no method has yet been found for determining the arithmetic nature of even beta values (similarly to the Riemann zeta function at odd integers greater than 3). The number $\backslash beta(2)=G$ is known as Catalan's constant.

It has been proven that infinitely many numbers of the form $\backslash beta(2n)${{Cite journal |last1=Rivoal |first1=T. |last2=Zudilin |first2=W. |date=2003-08-01 |title=Diophantine properties of numbers related to Catalan's constant |url=https://link.springer.com/article/10.1007/s00208-003-0420-2 |journal=Mathematische Annalen |language=en |volume=326 |issue=4 |pages=705–721 |doi=10.1007/s00208-003-0420-2 |issn=1432-1807|hdl=1959.13/803688 |hdl-access=free }} and at least one of the numbers $\backslash beta(2),\; \backslash beta(4),\; \backslash beta(6),\; ...,\; \backslash beta(12)$ are irrational.{{Cite journal |last=Zudilin |first=Wadim |date=2019-05-31 |title=Arithmetic of Catalan's constant and its relatives |journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg |volume=89 |issue=1 |pages=45–53 |doi=10.1007/s12188-019-00203-w |issn=0025-5858|doi-access=free |arxiv=1804.09922 }}

The even beta values may be given in terms of the polygamma functions and the Bernoulli numbers:{{Cite journal |last=Kölbig |first=K. S. |date=1996-11-12 |title=The polygamma function ψ(k)(x) for x=14 and x=34 |url=https://www.sciencedirect.com/science/article/pii/S0377042796000556 |journal=Journal of Computational and Applied Mathematics |volume=75 |issue=1 |pages=43–46 |doi=10.1016/S0377-0427(96)00055-6 |issn=0377-0427}}

:$$

\beta(2n)=\frac{\psi^{(2n-1)}(1/4)}{4^{2n-1}(2n)!}n - \frac{\pi^{2n}(2^{2n}-1)|B_{2n}|}{2(2n)!}

We can also express the beta function for positive $n$ in terms of the inverse tangent integral:

:$\backslash beta(n)=\backslash text\{Ti\}\_n(1)$

:$\backslash beta(1)=\backslash arctan(1)$

For every positive integer k:{{citation needed|date=September 2016}}

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

where $A\_\{k\}$ is the Euler zigzag number.

For negative odd integers, the function is zero:

:$\backslash beta(-2n-1)\backslash ;=\backslash ;0$

For every negative even integer it holds:

:$\backslash beta(-2n)\backslash ;=\backslash ;\backslash frac12E\_\{2n\}$.

It further is:

:$\backslash beta(0)\backslash ;=\backslash ;\; \backslash frac\{1\}\{2\}$.

We have:

$\backslash beta\text{'}(-1)=\backslash frac\{2G\}\backslash pi$

$\backslash beta\text{'}(0)=2\backslash ln\backslash Gamma(\backslash tfrac14)-\backslash ln\backslash pi-\backslash tfrac32\backslash ln2$

\beta'(1)=\tfrac\pi4(\gamma+2\ln2+3\ln\pi-4\ln\Gamma(\tfrac14))

with $\backslash gamma$

being Euler's constant and $G$

being Catalan's constant. The last identity was derived by Malmsten in 1842.{{Cite journal |last=Blagouchine |first=Iaroslav V. |date=2014-10-01 |title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results |url=https://link.springer.com/article/10.1007/s11139-013-9528-5 |journal=The Ramanujan Journal |language=en |volume=35 |issue=1 |pages=21–110 |doi=10.1007/s11139-013-9528-5 |issn=1572-9303|url-access=subscription }}

• Hurwitz zeta function
• Dirichlet eta function
• Polylogarithm

{{Reflist}}

• {{cite journal

|first1=M. L.

|last1=Glasser

|title=The evaluation of lattice sums. I. Analytic procedures

|year=1972

|journal=J. Math. Phys.

|doi=10.1063/1.1666331

|volume=14

|issue=3

|page=409

|bibcode=1973JMP....14..409G

}}

• J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.

Category:Zeta and L-functions