# Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?

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 ?

• Why the downvotes ? Because this is certainly a research level question ! – OneTwoOne Dec 23 '18 at 7:17
• 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 $\int_{\Re(s)=1/2} \omega$ – reuns Dec 24 '18 at 3:00