# Zeta function in the critical strip

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.

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

Use the dirichlet eta function relation.

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

in 0< re s <1 ?

yes.

or, is there a way to calculate values in the critical strip with out using an alternating series?

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.

Last edited:
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?

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($$\Re s > 0, s \not= 1$$)

Thanks!!