Bombieri's refinement

Bombieri [1 841 692]has given the following version of Weil's criterion: The Riemann Hypothesis holds if and only if

$$\sum_{\rho}\hat{g}(\rho) \hat{\overline{g}}(1-\rho)>0$$

for every complex-valued $g(x)\in C_0^\infty(0,\infty)$ which is not identically 0, where

$$\hat{g}(s)=\int_0^\infty g(x)x^{s-1}~dx$$

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