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

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SUMMARY

The value of the Riemann Zeta Function at 2 is established as \(\zeta(2) = \frac{\pi^2}{6}\), which is derived from the infinite series \(\sum^{\infty}_{n=1}\frac{1}{n^2}\). This proof can be accessed through various resources, including an elementary proof on Wikipedia and Euler's original derivation found in the American Mathematical Society's bulletin. Fourier series methods provide shorter and more generalized proofs, and Xuming Chen's paper in the College Mathematics Journal offers recursive formulas for \(\zeta(2k)\) and \(L(2k - 1)\).

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  • Understanding of geometric series and convergence
  • Familiarity with the Riemann Zeta Function
  • Basic knowledge of Fourier series
  • Ability to interpret mathematical proofs and derivations
NEXT STEPS
  • Study the elementary proof of \(\zeta(2)\) on Wikipedia
  • Explore Euler's original derivation of the Zeta function values
  • Research Fourier series methods for proving Zeta function values
  • Read Xuming Chen's paper on recursive formulas for \(\zeta(2k)\) and \(L(2k - 1)\)
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Mathematics students, educators, and researchers interested in number theory, particularly those studying the properties and proofs related to the Riemann Zeta Function.

imranq
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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!
 
Last edited by a moderator:

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