Talk:Dirichlet beta function

Add topic
Active discussions
WikiProject Mathematics (Rated Start-class, Low-priority)
WikiProject iconThis article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Start This article has been rated as Start-Class on the project's quality scale.
 Low  This article has been rated as Low-priority on the project's priority scale.
 

Reciprocal of this function - is this worth adding to the main section?Edit

It may be worth mentioniong that:

   1/beta(s) = sum(n=1..infinity((-1)^n * mu(n+1)/(2*n+1)^s));

which is valid for Re(s)>1. Hair Commodore (talk) 20:05, 28 July 2008 (UTC). Note that mu(*) is the Mobius mu function. (It certainly converges to 4/Pi when s=1)

This formula is wrong! I tried checking it for 's'=1, 's'=2, and 's'=3, and it works at none of them! 81.102.15.200 (talk) 13:36, 18 August 2008 (UTC)

Apologies - I should have said that:

   1/beta(s) = sum(n=1..infinity,(((-1)^n * mu(2*n+1))/(2*n+1)^s))

in Maple notation - rather than what I typed previously.The anaonymous user who questioned it was quite correct. I made a wally error, so I have now corrected it. (It is still correct for 'Re'(s)>1) Hair Commodore (talk) 19:24, 18 August 2008 (UTC)

yes, from the Euler product I just added :
 
I'd still like to know if the Riemann hypothesis for   would imply (or be implied by) the one for  , or if the two are completely independent. 78.227.78.135 (talk) 17:10, 4 January 2016 (UTC)

DefinitionEdit

The formula for beta(s) in terms of the polygamma function only seems to be valid at even positive integers. I believe multiplying this formula by (-1)^s makes it valid at odd positive integers as well as even positive integers. The conditions for the validity of the polygamma formula for beta(s) need to be specified or the reader may assume the formula is valid for all complex values of s similar to the two preceding formulas which express beta(s) in terms of the Hurwitz zeta function and the Lerch transcendent.StvC (talk) 16:22, 2 April 2021 (UTC)

I corrected and clarified the polygamma formula.StvC (talk) 17:06, 4 April 2021 (UTC)