Lindelöf Hypothesis

Establish that |ζ(1/2 + it)| ≪_ε t^ε for every ε > 0, equivalently proving that the growth exponent μ(1/2) equals 0.

Background

The Lindelöf Hypothesis provides a key upper bound for the size of ζ(1/2+it) and has profound implications for moments and subconvexity problems.

The paper reviews standard implications of the Lindelöf Hypothesis via the convexity of the growth exponent μ(σ) and the functional equation.

References

The Lindelöf Hypothesis states that $\vert \zeta(1/2 + it) \vert \ll t{\epsilon}$ for any $\epsilon > 0$.

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