What is the proof for the value of the Riemann Zeta Function at 2?

AI Thread Summary
The discussion centers on the value of the Riemann Zeta Function at 2, specifically that ζ(2) equals π²/6, which is derived from the sum of the infinite series of reciprocals of squares. Participants express interest in accessible proofs, noting that Fourier analysis provides a clearer understanding, despite initial confusion with vague explanations. References to Euler's original derivation and various resources, including Wikipedia and academic papers, are shared to aid comprehension. Additionally, the conversation highlights the connection between zeta function values and Bernoulli numbers through Fourier series. Overall, the exchange emphasizes the importance of different mathematical approaches to understanding this significant result.
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So we were going over geometric series in my calc class (basic, I know), however I was intrigued by one point that the prof. made during lecture

\frac{\pi^2}{6} = \sum^{\infty}_{n=1}\frac{1}{n^2} = \zeta (2)

That's amazing (at least to me). Looking for the explanation for this, I found a bunch of stuff relating to Fourier analysis which was - unfortunately - written in vague terms. Would someone explain this proof that is accessible to a Calc II student? Thanks
 
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That was fast and I understand it now, thanks! By the way, what is the Fourier series used for?
 
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you might enjoy the computation in euler's precalculus book of this series. he also did it for many other values of zeta, and apparently gave a general formula relating all even values of zeta to the bernoulli numbers, as given in the appendix to milnor and stasheff, characteristic classes, using Fourier series.
 
http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf
Gives 14 different evaluations (its also linked on the wikipedia page incidentally).

The part on zeta function values at http://www.ams.org/bull/2007-44-04/S0273-0979-07-01175-5/S0273-0979-07-01175-5.pdf, guides you through Euler's original derivation (which is easily made rigorous once the infinite product for the sine is obtained).

Fourier series methods offer shorter and easily generalised proofs, for instance, check the following threads,

https://www.physicsforums.com/showthread.php?t=192765
and
https://www.physicsforums.com/showthread.php?t=95994

There was also an elementary paper by Xuming Chen in the College Mathematics Journal called 'Recursive Formulas for \zeta(2k)[/tex] and L(2k - 1)[/tex]'.<br /> <br /> Hope this helped!
 
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yasiru89 said:
http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf
Gives 14 different evaluations (its also linked on the wikipedia page incidentally).

The part on zeta function values at http://www.ams.org/bull/2007-44-04/S0273-0979-07-01175-5/S0273-0979-07-01175-5.pdf, guides you through Euler's original derivation (which is easily made rigorous once the infinite product for the sine is obtained).

Fourier series methods offer shorter and easily generalised proofs, for instance, check the following threads,

https://www.physicsforums.com/showthread.php?t=192765
and
https://www.physicsforums.com/showthread.php?t=95994

There was also an elementary paper by Xuming Chen in the College Mathematics Journal called 'Recursive Formulas for \zeta(2k)[/tex] and L(2k - 1)[/tex]'.<br /> <br /> Hope this helped!
<br /> <br /> Thanks a lot!
 
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