Question about Riemann Zeta Function

  • Thread starter willr12
  • Start date
  • #1
17
2
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....
 

Answers and Replies

  • #2
37
0
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:
  • #3
Khashishi
Science Advisor
2,815
493
I'd say, calculate the first m terms, and the first m+1, m+2... and see what it converges to.
 

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