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Eugine_Nier comments on The Use of Many Independent Lines of Evidence: The Basel Problem - Less Wrong
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An interesting thing about the GRH is that at oft neglected piece of evidence against it is how the Siegal zero seems to behave like a real object, i.e., having consistent properties.
I'm not sure what you mean here. If it didn't have consistent properties we could show it doesn't exist. Everything looks consistent up until the point you show it isn't real. Do you mean that it has properties that don't look that implausible? That seems like a different argument.
If we can easily prove a conjecture except for some seemingly arbitrary case, that's evidence for the conjecture being false in that case.