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When I speak about intuition behind the proof of the functional equation. I am talking about proofs similiar to this one http://www.math.harvard.edu/~elkies/M259.02/zeta1.pdf

As far as I can tell, this idea was originally formulated out of necessity. Reimann Riemann needed it for the prime number theorem. However the intuition becomes more natural ifaccepts if one accepts both these of the following facts:

1)The Cahen Mellin Integral transforms a dirichlet Dirichlet series (something hard to work with) into a fourier Fourier series/polynomial (which we usually are more familiar with).

2) When applying $\zeta(2s)$ to this transform, we get the jacobi Jacobi theta function $\theta(y)$ which has tons of special structure to it.

So in a sense it is the classic scheme of "I can't work with A so I will transform it to B and then transform it back." When you transform the $\zeta(2s)$ you are getting a fourier Fourier series (which by itself has a ton of facts about it) but it also has a really useful functional equation to it.
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When I speak about intuition behind the proof of the functional equation. I am talking about proofs similiar to this one http://www.math.harvard.edu/~elkies/M259.02/zeta1.pdf

As far as I can tell, this idea was originally formulated out of necessity. Reimann needed it for the prime number theorem. However the intuition becomes more natural if one accepts two ifaccepts both these facts:

1)The Cahen Mellin Integral transforms a dirichlet series (something hard to work with) into a fourier series/polynomial (which we usually are more familiar with).

2) When applying $\zeta(2s)$ to this transform, we get the jacobi theta function $\theta(y)$ which has tons of special structure to it.

So in a sense it is the classic scheme of "I can't work with A so I will transform it to B and then transform it back." When you transform the $\zeta(2s)$ you are getting a fourier series (which by itself has a ton of facts about it) but it also has a really useful functional equation to it.
show/hide this revision's text 1

When I speak about intuition behind the proof of the functional equation. I am talking about proofs similiar to this one http://www.math.harvard.edu/~elkies/M259.02/zeta1.pdf

As far as I can tell, this idea was originally formulated out of necessity. Reimann needed it for the prime number theorem. However the intuition becomes more natural if one accepts two facts:

1)The Cahen Mellin Integral transforms a dirichlet series (something hard to work with) into a fourier series/polynomial (which we usually are more familiar with).

2) When applying $\zeta(2s)$ to this transform, we get the jacobi theta function $\theta(y)$ which has tons of special structure to it.

So in a sense it is the classic scheme of "I can't work with A so I will transform it to B and then transform it back." When you transform the $\zeta(2s)$ you are getting a fourier series (which by itself has a ton of facts about it) but it also has a really useful functional equation to it.