The question is in the title : suppose RH is false. Would it imply that there exists a yet to be discovered functional equation relating the values of the Riemann zeta function at different points themselves related in some specific way ? As far as I know, the functional equation as we know it is required as a necessary condition for RH to be true, but not a sufficient one. Are references on this subject available ?
2026-03-25 22:05:07.1774476307
Would the falsity of RH imply the existence of a yet unknown functional equation?
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No.
This is an odd statement. There are plenty of functions that have constrained zeroes but no functional equation. So this is a bit of a mystery.
However, you may be interested to know that there other functions $L(s)$ which are similar to the Riemann zeta function: $L(s) = \sum a(n) n^{-s}$ and $L(s)$ has a functional equation of the shape $L(s) \sim L(1-s)$, just like $\zeta(s)$. But some of these functions have "nontrivial" zeroes off the line $\mathrm{Re} (s) = \frac{1}{2}$. So merely being a Dirichlet series with functional equation is not enough.
[These functions $L(s)$ that I am referring to are the $L$-functions associated to modular forms of half-integral weight, $L(s,f)$. Note that these do not have an Euler product, so there is something a bit different going on.]