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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?

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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).

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    $\begingroup$ That book has so many great things in it! $\endgroup$
    – Rob Harron
    Commented Apr 26, 2011 at 4:51
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    $\begingroup$ you can also see my paper :) vixra.org/pdf/1310.0048v4.pdf $\endgroup$
    – user23964
    Commented Aug 25, 2019 at 11:47

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