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We can derive from the Poisson summation formula the modularity of the Theta function, which results in the functional equation. In his book on the Riemann Zeta function, Patterson mentions also that there is a way back. How can this be done?

Equivalently, how can the modularity of the Theta function (which is Poisson summation for a one parameter family of functions only) be used to derive the Poisson summation formula for general test functions?

edited Mar 29 2012 at 9:20

asked Apr 25 2011 at 21:00

Marc Palm
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Daniel Parry · Accepted Answer · 2011-04-26 04:13:24Z UTC

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The theorem you are looking for is Theorem 10.2.17 in Henri Cohen's book Number Theory: Analytic and Modern Tools (Google Books Link).

answered Apr 26 2011 at 4:13

Daniel Parry
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That book has so many great things in it! – Rob Harron Apr 26 2011 at 4:51

Why is the functional equation of the Riemann zeta function equivalent to the Poisson summation formula?

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