The Lindelöf Hypothesis

Claim: \(\zeta\left(\tfrac12 + it\right) = O(t^{\varepsilon})\) for every \(\varepsilon > 0\). It follows from RH but is strictly weaker — potentially provable by other means.

A century of exponent-chasing: from Lindelöf's own \(t^{1/2}\)-scale convexity through Walfisz, Huxley's \(t^{131/416+ε}\) long reign, to Bourgain's 2017 landmark \(t^{13/84+ε}\) ≈ \(t^{0.1548+ε}\) via decoupling technology imported from PDE. Each improvement reshapes zero-density estimates and hence primes-in-short-intervals. The hypothesis controls the distribution of ζ's large values; its resolution — even by RH-independent means — would be a structural event in analytic number theory.