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Question about Riemann Zeta Function

  1. Dec 1, 2014 #1
    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. jcsd
  3. Dec 1, 2014 #2
    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
  4. Dec 4, 2014 #3
    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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