A nice result of Balazard et al says the Riemann Hypothesis is equivalent to the statement that

$$\int_{-\infty}^{\infty} \frac{\log|\zeta(1/2 + it)|}{\frac{1}{4}+t^2} \mathrm{d}t=0$$ where $\zeta$ is the zeta function of Riemann.

Is there an analogue of this criterion for the Generalised Riemann Hypothesis ? My classmates aren't optimistic that it exists, because it seems Balazard et al's proof relies crucially on the fact that $\lim_{s\rightarrow 1}((s-1)\zeta(s))=1$, which is not true for other L-functions.

  • 2
    $\begingroup$ It sure has a counterpart for all Dedekind zeta functions. Have you looked at papers by J.-F. Burnol, such as An adelic causality problem related to abelian $L$-functions? Or, by "other $L$-functions," do you intend automorphic $L$-functions in a sweeping generality? $\endgroup$ – Vesselin Dimitrov Nov 24 '18 at 12:06
  • $\begingroup$ @VesselinDimitrov, i wasn't aware of those papers. Let me search them on Google. Thanks. By ''L-functions, i indeed meant $L(s, \chi)$ for all $\chi$...but it would be quite interesting if there is any other L-function for which the analogue holds. $\endgroup$ – EinsteinDriveResident Nov 24 '18 at 12:09
  • $\begingroup$ I'm voting to close this question as off-topic because the person behind these different user accounts doesn't seem to have come back to this one and does not seem interested in following up on this particular question with this particular account $\endgroup$ – Yemon Choi Dec 20 '18 at 8:25
  • $\begingroup$ Have you managed to discuss this with your "classmates" yet? $\endgroup$ – Yemon Choi Dec 23 '18 at 20:54
  • $\begingroup$ @YemonChoi When you say "these different user accounts", are you referring to other questions? If so, do you still (3 months later) have links to those questions? $\endgroup$ – user44191 Mar 14 at 0:25

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