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    $\begingroup$ Dear Terry, I'm curious as to why you don't think the Selberg class does the job of axiomatizing the correct family of L-functions (for RH etc.). Regards, Matthew $\endgroup$ Commented Oct 25, 2012 at 19:02
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    $\begingroup$ Huh, it appears my information here is out of date (I knew about RH counterexamples to previous axiomatisations of L-functions, but didn't realise that Selberg's formulation manages to avoid all known counterexamples.) I'll update the text accordingly. (I'm still skeptical that these sorts of axioms aren't just abstracting what we already know how to do with L-functions, rather than being the way forward to discover new proof methods that might make progress on problems such as RH, but I would of course be very happy to be proven incorrect on this point.) $\endgroup$ Commented Oct 25, 2012 at 21:19
  • $\begingroup$ Selberg's conjectures are very interesting! Thank you, Matt and Terry, for bringing those up. $\endgroup$ Commented Oct 26, 2012 at 1:01