Open problem, Arithmetic and number theory, Analysis, posed 1908

Lindelöf hypothesis

Open

For every $\varepsilon > 0$ , the Riemann zeta function on the critical line satisfies $\zeta\!\left(\tfrac{1}{2} + it\right) = O(t^{\varepsilon})$ as $t \to \infty$ ; equivalently, the growth exponent $\mu(\sigma) = \inf\{a : \zeta(\sigma+it) = O(t^a)\}$ satisfies $\mu\!\left(\tfrac{1}{2}\right) = 0$ .

Research frontier as of 2026

As of 2026, the Lindelöf hypothesis remains open. The best published unconditional pointwise bound on the critical line is Bourgain's 2017 estimate $\mu(\tfrac{1}{2}) \le \tfrac{13}{84} \approx 0.15476$ , obtained by injecting Bourgain–Demeter $\ell^2$ decoupling into the Bombieri–Iwaniec method (improving Huxley's $\tfrac{32}{205} \approx 0.15610$ ). For moments of $\zeta(\tfrac{1}{2}+it)$ , asymptotic formulas are known unconditionally for the second ( $2k=2$ ) and fourth ( $2k=4$ ) moments, while even the sixth moment ( $2k=6$ ) remains open. On the zero-density consequence of the Lindelöf hypothesis (the density hypothesis $N(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon}$ ), Guth and Maynard achieved a landmark breakthrough in 2024, breaking Ingham's 1940 barrier at $\sigma = \tfrac{3}{4}$ with $N(\sigma, T) \ll T^{\frac{30}{13}(1-\sigma)+o(1)}$ .

Best known results

Unconditional pointwise bound: $\zeta(\tfrac{1}{2}+it) \ll t^{13/84+\varepsilon}$ with $\tfrac{13}{84} \approx 0.15476$ (Bourgain, 2017).
Conditional bound under the Riemann hypothesis: $|\zeta(\tfrac{1}{2}+it)| \ll \exp\!\left(C \frac{\log t}{\log\log t}\right)$ (Littlewood, 1924).
Zero-density bound toward the density hypothesis: $N(\sigma, T) \ll T^{\frac{30}{13}(1-\sigma)+o(1)}$ (Guth and Maynard, 2024).

Tools and where they stop

Tool	Achieved	Where it stops
Bombieri–Iwaniec method and $\ell^2$ decoupling	Breaks the approximate functional equation into short exponential sums approximated by rational phases and bounds the resonance sets via harmonic analysis decoupling, reaching $\mu(\tfrac{1}{2}) \le \tfrac{13}{84}$ .	Local Taylor approximations of $t \log n$ only capture cancellation on short intervals of $n$ , leaving a positive exponent barrier well above $0$ .
Large-value estimates for Dirichlet polynomials (Guth–Maynard method)	Controls how often a Dirichlet polynomial $\sum_{n \sim N} a_n n^{-it}$ can be large by analyzing singular values of raised time-frequency matrices, proving $N(\sigma, T) \ll T^{\frac{30}{13}(1-\sigma)+o(1)}$ .	Controls average zero density in the strip $\sigma > \tfrac{1}{2}$ rather than pointwise sixth-and-higher moments on the critical line $\sigma = \tfrac{1}{2}$ .

Open questions

Can the sixth moment $\int_0^T |\zeta(\tfrac{1}{2}+it)|^6\,dt \ll T^{1+\varepsilon}$ be proved unconditionally?
Can the unconditional density hypothesis $N(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon}$ be proved for all $\tfrac{1}{2} \le \sigma \le 1$ without first proving the Lindelöf hypothesis?

References

Ernst Lindelöf (1908). Quelques remarques sur la croissance de la fonction ζ(s)
E. C. Titchmarsh (1986). The Theory of the Riemann Zeta-Function (2nd ed., rev. by D. R. Heath-Brown)
Jean Bourgain (2017). Decoupling, exponential sums and the Riemann zeta function · DOI:10.1090/jams/860 · arXiv:1408.5794
Larry Guth, James Maynard (2024). New large value estimates for Dirichlet polynomials · arXiv:2405.20552