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The Balazard-Saias-Yor criterion for the Riemann Hypothesis states that the latter is equivalent to the statement that

$$\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s^2|}|ds|=0$$ where $\zeta$ denotes the Riemann zeta function. But does this criterion also hold if $\zeta$ is the Weil zeta function ?

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  • $\begingroup$ Why the downvotes ? Because this is certainly a research level question ! $\endgroup$
    – OneTwoOne
    Commented Dec 23, 2018 at 7:17
  • $\begingroup$ In Balazard's paper they make it clear that $\int_{\Re(s)=1/2} \frac{\log|F(s)|}{s (1-s)}|ds|=2\pi \sum_{\Re(\rho) > 1/2} \log |\rho| - \log |1-\rho|$ for a wide class of analytic functions $F(s)$. iml.univ-mrs.fr/~balazard/pdfdjvu/9.pdf Note if $F(s)$ is analytic and $F(\overline{s}) = \overline{F(s)}$ then $\omega = \frac{\log F(s)}{s(1-s)}ds-\frac{\log F(\overline{s})}{\overline{s}(1-\overline{s})}d\overline{s}$ is an harmonic form away from $s=1$ and the zeros of $F(s)$ so we can try applying the residue theorem to $\int_{\Re(s)=1/2} \omega$ $\endgroup$
    – reuns
    Commented Dec 24, 2018 at 3:00

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There is no Weil Zeta Function per se, but a bunch of such associated with various algebraic-geometric objects (the simplest ones are associated with elliptic curve structures on toruses); such functions are rational functions (very unlike the highly transcendent classical zeta) and the analogue of the Riemann Hypothesis there is a statement about the decomposition in factors of the numerators of such and it's an algebraic-combinatorial statement essentially, not an analytic one like in the classical case.

The RZ book by Patterson explains how the same combinatorial statement extrapolated for the complex field is simply false but that has nothing really to do with the Riemann Hypothesis per se.

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