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This may be a stupid question but is there a way to calculate Riemann's Zeta Function by hand exactly or can you only estimate it?

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You can certainly compute the values at the even integers "by hand." Which values do you want? –  Potato Apr 10 '13 at 1:45
    
As simple as I can get. Just trying it out. –  The Diamante Apr 10 '13 at 1:48
    
Well, then it was no joke: Potato already told you that the values of zeta at the even ( negative, I'm guessing he meant) integers are pretty easy to check by hand...even sleeping, in the dark and dancing rock'n'roll –  DonAntonio Apr 10 '13 at 1:52
    
Thank you! How would you go about doing this though? –  The Diamante Apr 10 '13 at 1:54
    
This question seems to be relevant math.stackexchange.com/questions/3271/… –  Baby Dragon Apr 10 '13 at 1:55
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2 Answers

The functional equation of the zeta function gives you the trivial zeros at once:

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

if you input now $\,s=-2k\;,\;\;k\in\Bbb N\,$ , you get

$$\zeta(-2k)=2^{-2k}\pi^{-2k-1}\sin\left(\frac{\pi (-2k)}{2}\right)\Gamma(1+2k)\zeta(1+2k)=0\ldots$$

Question: Why in the negative even integers and not in the positive ones? Hint: check stuff about the gamma function.

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I have found many questions and pages in the Gamma Function and understand the basics of it but can not find what the individual variables represent/their values. –  The Diamante Apr 10 '13 at 2:34
    
Ok @TheDiamante, then you probably know the Gamma function has poles on the negative integers, and that's why in the above relation we had to take even negative integers... –  DonAntonio Apr 10 '13 at 2:53
    
Yes, I understand this but I do not understand how to find all the variable of the Gamma Function. How would this be achieved? –  The Diamante Apr 10 '13 at 3:05
    
I don't understand what you mean by "all the variable of the Gamma Function"... –  DonAntonio Apr 10 '13 at 3:06
    
There are variables within the function that I can't find how to find their values. As here en.wikipedia.org/wiki/Gamma_function, I don't know the values of the variables d, e and t in the second formula. –  The Diamante Apr 10 '13 at 3:11
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Riemann did it - that's how he came up with his famous hypotheses. Of course, it was very arduous and he had to derive some pioneering results to do it, but that's why he is famous and I am not.

Do a search for "computing the Riemann zeta function" and you will get many hits. A good reference is Edwards' book "Riemann's Zeta Function".

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(Note this answer is concerning the numerical approximation of the zeta function.) –  anon Apr 10 '13 at 4:10
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