Does the functional equation of the Selberg Zeta function imply the Selberg trace formula?
BTW, the trace formula implies the functional equation.
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$\begingroup$ Is that really ture ? $\endgroup$– Chandan Singh DalawatCommented Oct 18, 2012 at 4:14
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$\begingroup$ I am sure it is ture. By taking some special test function in the trace formula we get the functional equation of the Selberg Zeta function. $\endgroup$– 7-adicCommented Oct 18, 2012 at 5:37
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No, you also need that there aren't too many zeros.
Then it works. I indicated the strategy here The Guinand-Weil explicit formula without entire function theory just in terms of the Riemann Zeta function.
Also there is a Lecture Notes in Mathematics by Jürgen Fischer (PhD thesis).
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$\begingroup$ Note the fact, that there aren't too many zeros can be read off from the functional equation in principle, since it together with the ''Euler product'' implies that the Selberg zeta function is of finite exponential order. $\endgroup$ Commented Oct 18, 2012 at 11:04