Existence of a p-adic analogue of the Riemann Hypothesis

Ascertain whether a meaningful analogue of the Riemann Hypothesis exists for p-adic zeta and L-functions, and if so formulate and prove such a statement.

Background

While p-adic zeta and L-functions are fundamental in Iwasawa theory and arithmetic geometry, no direct analogue of RH is currently known for them.

The paper notes the absence of such an analogue despite the rich theory surrounding p-adic L-functions.

References

For completeness, I will also touch upon $p$-adic functions: while no analogue of the Riemann Hypothesis is known in these cases, they have nonetheless given rise to a wealth of beautiful developments.

— The Riemann Hypothesis: Past, Present and a Letter Through Time  (2602.04022 - Connes, 3 Feb 2026) in Introduction