# Question about Riemann Zeta Function

1. Dec 1, 2014

### willr12

I understand how to calculate values of positive values ζ(s), it's pretty straightforward convergence. But when you expand s into the complex plane, like ζ(δ+bi), how do you assign a value with i as an exponent? Take for example
ζ(1/2 + i)
This is the sequence
1/1^(1/2+i) + 1/2^(1/2+i) + 1/3^(1/2+i) .......
How do you assign a value to this? do you have to use euler's identity to calculate it? Or am I looking at it all wrong?
P. S. Try to dumb it down for me. I'm in algebra 2 right now....

2. Dec 1, 2014

### AMenendez

First, Euler's Identity has nothing to do with the Reimann-Zeta function. Euler's Identity is just
$e^{i \pi} + 1 = 0$
Euler's Formula is probably what you're thinking of (which is where the above identity comes from):
$e^{ix} = \cos x + i \sin x$
But there's no exponential function (it works that way because the exponential function is re-written with a power series to get the above formula), and $i$ would have to be factored out of the exponent entirely to rewrite it that way (it's only in one of the terms).
But, in the case you're talkng about where $s$ is defined as $\sigma + bi$ for $0 < \sigma < 1$, then it can be left as is, but the imaginary zeroes are plotted along the "critical line", $\Re(s) = \frac{1}{2}$

http://mathworld.wolfram.com/RiemannZetaFunction.html

Last edited: Dec 1, 2014
3. Dec 4, 2014

### Khashishi

I'd say, calculate the first m terms, and the first m+1, m+2... and see what it converges to.