Let zeta(s) denote the Riemann zeta function, i.e. the meromorphic continuation to Csetminus1 of the function defined by zeta(s)=∑_n=1^∞ n^-s for operatornameRe(s)>1. For sigma∈R define mu(sigma)=inf,a∈R;:;zeta(sigma+iT)=O(T^a) as T→∞,, where the O-estimate is taken as the real variable T tends to +∞. Then mu!(tfrac12)=0, that is, for every varepsilon>0 there is a constant C(varepsilon)>0 such that |zeta!(tfrac12+it)|≤ C(varepsilon),t^varepsilonqquadfor all t≥ 2 . Equivalently, zeta!(tfrac12+it)=o(t^varepsilon) as t→∞ for every varepsilon>0.

Lindelöf hypothesis

Let $\zeta(s)$ denote the Riemann zeta function, i.e. the meromorphic continuation to $\mathbb{C}\setminus\{1\}$ of the function defined by $\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$ for $\operatorname{Re}(s)>1$ .

For $\sigma\in\mathbb{R}$ define

\mu(\sigma)=\inf\{\,a\in\mathbb{R}\;:\;\zeta(\sigma+iT)=O(T^{a})\ \text{as } T\to\infty\,\},

where the $O$ -estimate is taken as the real variable $T$ tends to $+\infty$ .

Then

\mu\!\left(\tfrac{1}{2}\right)=0,

that is, for every $\varepsilon>0$ there is a constant $C(\varepsilon)>0$ such that

\left|\zeta\!\left(\tfrac{1}{2}+it\right)\right|\le C(\varepsilon)\,t^{\varepsilon}\qquad\text{for all } t\ge 2 .

Equivalently, $\zeta\!\left(\tfrac12+it\right)=o(t^{\varepsilon})$ as $t\to\infty$ for every $\varepsilon>0$ .

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The Lindelöf Hypothesis
Let $Z(t)$ denote the real-valued Hardy function associated with the Riemann zeta function, and let $\varepsilon>0$ .

Lindelöf Hypothesis. As $t\to\infty$ ,
$Z(t)=O_{\varepsilon}(t^{\varepsilon}).$
Equivalently,
$Z(t)=e^{o(\log t)}.$
The Lindelöf Hypothesis is a central conjecture about the growth of the zeta function on the critical line and is weaker than the Riemann Hypothesis. Its status is open.
source: David W. Farmer, “Currently there are no reasons to doubt the Riemann Hypothesis: The zeta function beyond the realm of computation”, arXiv:2211.11671 (2025).

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Lindelöf hypothesis

Equivalent formulations 1

The Lindelöf Hypothesis

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