Zeta function in the critical strip

[TheOogy]

how do i calculate values of the riemann zeta function in the critical strip? because if you only know zeta as a series:

$$\zeta(s) = \sum 1/n^s$$

and the functional equation

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

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.

 

[rscosa]

Hi!,
there are many other representations (wikipedia or www.mathworld.com) but maybe non of them will be enough helpfull.

 

[Santa1]

Use the dirichlet eta function relation.

 

[TheOogy]

can we express the eta function as a product of primes?

 

[Santa1]

in 0< re s <1 ?

 

[TheOogy]

yes.

 

[TheOogy]

or, is there a way to calculate values in the critical strip without using an alternating series?

 

[Santa1]

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 $$\eta(s)\Gamma(s)$$ as an integral,

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

and then use the zeta relation again.

You could also use the $$\zeta(s)\Gamma(s)$$ integral form, and deform the contour as riemann originally did.

 

[TheOogy]

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?

 

[Santa1]

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,

$$\zeta(s) = \frac{1}{(1-2^{1-s})\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x+1}\mathrm{d}x$$, edit($$\Re s > 0, s \not= 1$$)

 

[TheOogy]

Thanks!