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Let us suppose we got a modification of the Selberg trace as follows

$$ \sum_{n=0}^{\infty} h(r_n) = \frac{\mu(F)}{4 \pi } \int_{-\infty}^{\infty} r \, h(r) \tanh(\pi r) dr + \sum_{ \{T\} } \frac{ \log N(T_0) }{ N(T)^{1/2} - N(T)^{-1/2} } g \left ( \log N(T) \right )+h(i/2)- \sum_{n=1}^{\infty} \frac{\Lambda (n)}{\sqrt{n}}g(logn)- \frac{g(0)log\pi}{2}+ \frac{1}{4\pi}\int_{-\infty}^{\infty}drh(r) \frac{ \Gamma '}{\Gamma}(1/4+ir/2)$$

so the trace over the riemann zeros is included inside this model, would this trace (the existence) be a proof of the Riemann Hypothesis?

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up vote 1 down vote accepted

What quotient are you working on? I guess $SL_2(\mathbb{Z}) \backslash \mathbb{H}$, right?

Okay, I guess you have hidden the contribution of the continuous spectrum, which looks roughly like $$ \int\limits_{\Re s} h(is) \cdot d/dx \log \frac{\zeta(2s+1)}{\zeta(2s)} d s,$$ in terms of the zeros of $\zeta$ as done in derivation of the Weil explicit formula?

But you don't get quite, what you write. The zeros of $\zeta$ in turn up here at $\Re s=3/4$ assuming RH (note the scaling).

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