Zeta function in the critical strip

  • Thread starter TheOogy
  • Start date
  • #1
16
0
how do i calculate values of the riemann zeta function in the critical strip? because if you only know zeta as a series:

[tex]
\zeta(s) = \sum 1/n^s
[/tex]

and the functional equation

[tex]
\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s) \!
[/tex]

you can only calculate values that have real part bigger then 1 or smaller then 0.
i know i can use a math software to calculate it but i want to understand the process.
 

Answers and Replies

  • #2
14
0
Hi!,
there are many other representations (wikipedia or www.mathworld.com) but maybe non of them will be enough helpfull.
 
  • #3
107
0
Use the dirichlet eta function relation.
 
  • #4
16
0
can we express the eta function as a product of primes?
 
  • #5
107
0
in 0< re s <1 ?
 
  • #6
16
0
yes.
 
  • #7
16
0
or, is there a way to calculate values in the critical strip with out using an alternating series?
 
  • #8
107
0
Well, you can use the relation to zeta and use its euler product. But I'm not sure as far as the convergence goes.

edit1: And yes, you can (amongst other ways) express [tex]\eta(s)\Gamma(s)[/tex] as an integral,

[tex]\eta(s)\Gamma(s)=\int_0^\infty \frac{x^{s-1}}{e^x+1}\mathrm{d}x[/tex], valid for re s > 0.

and then use the zeta relation again.

You could also use the [tex]\zeta(s)\Gamma(s)[/tex] integral form, and deform the contour as riemann originally did.
 
Last edited:
  • #9
16
0
i tried using the euler product but it didn't work, but thanks for the eta-gamma integral, can you show me the zeta-gamma integral two and save me the search?
 
  • #10
107
0
Just go to almost any gamma or zeta function online encyclopedia site for more info, but beware the original form only works for re s > 1 (the eta form works for re s>0), if you are not somewhat familiar with complex analysis you won't get much of it.

The eta gamma + relation gives,

[tex]\zeta(s) = \frac{1}{(1-2^{1-s})\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x+1}\mathrm{d}x[/tex], edit([tex]\Re s > 0, s \not= 1[/tex])
 
  • #11
16
0
Thanks!!
 

Related Threads on Zeta function in the critical strip

  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
2
Views
4K
Replies
2
Views
4K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
10
Views
8K
Top