Math Conjectures

Lindelöf Hypothesis

OPENMajorConjectureProposed 1908 · Full conjecture

Canonical statement

Let ζ(s)=∑n≥1n−s\zeta(s)=\sum_{n\ge1}n^{-s} for ℜs>1\Re s>1, and let ζ\zeta also denote its meromorphic continuation to C\mathbb C. For every ε>0\varepsilon>0,
ζ ⁣(12+it)=Oε ⁣((1+∣t∣)ε)as ∣t∣→∞. \zeta\!\left(\tfrac12+it\right) =O_\varepsilon\!\left((1+|t|)^\varepsilon\right) \quad\text{as }|t|\to\infty .
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Let \(\zeta(s)=\sum_{n\ge1}n^{-s}\) for \(\Re s>1\), and let \(\zeta\) also denote its meromorphic continuation to \(\mathbb C\). For every \(\varepsilon>0\),
\[
  \zeta\!\left(\tfrac12+it\right)
    =O_\varepsilon\!\left((1+|t|)^\varepsilon\right)
    \quad\text{as }|t|\to\infty .
\]
NT-AN-004:[Lindelof1908][Titchmarsh1986Zeta][Bourgain2017Zeta]
Number theoryAnalysis and functional analysis

Notes

The Lindelöf hypothesis, put forward by Lindelöf in 1908 [Lindelof1908], asserts that the Riemann zeta function grows more slowly than any positive power along the critical line: for every ε>0\varepsilon>0, ζ(12+it)=Oε((1+∣t∣)ε)\zeta(\tfrac12+it)=O_\varepsilon((1+|t|)^\varepsilon). It is a statement about the size of ζ\zeta rather than its zeros, with well-known consequences for moments of ζ\zeta and for primes in short intervals.

The Phragmén–Lindelöf convexity principle gives the exponent 1/4+ε1/4+\varepsilon on the critical line, and a century of subconvexity work, chronicled in Titchmarsh's treatise [Titchmarsh1986Zeta], has lowered it by small increments; Bourgain's decoupling method gives ζ(12+it)≪ε∣t∣13/84+ε\zeta(\tfrac12+it)\ll_\varepsilon|t|^{13/84+\varepsilon} [Bourgain2017Zeta]. The Riemann hypothesis implies the Lindelöf hypothesis, but no converse implication is known.

The distance from such exponents to the conjectured 00 remains vast: every known method loses a fixed power of ∣t∣|t|, and closing that gap is exactly what a resolution requires. The hypothesis is open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.