Lindelof (pointwise growth) exponent for the Riemann zeta function

Define the infimal exponent μ_ζ by μ_ζ := infBiglθ≥ 0: lvertζ(1/2+it)rvert≪_ε(1+lvert trvert)^θ+ε for all ε>0Bigr. We define C_62a := μ_ζ, the Lindelof (pointwise growth) exponent for ζ(1/2+it).

Current state (tracked automatically)

Best known bounds for constant 62a: 0 ≤ C ≤ 0.15476190476190477. Upper bound ([Bou2017], Bourgain's pointwise bound for (1/2+it)); lower bound (Trivial from the definition C_62a≥ 0).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 62a)	0 Trivial from the definition C_62a≥ 0.	Tao's optimization constants
Best upper bound with an explicit value (constant 62a)	0.15476190476190477 [Bou2017] — Bourgain's pointwise bound for (1/2+it).	Tao's optimization constants

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@misc{cairn-constant-62a-lindelof-pointwise-growth-exponent-for-the-riemann-zeta-function,
  title        = {Lindelof (pointwise growth) exponent for the Riemann zeta function},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-62a-lindelof-pointwise-growth-exponent-for-the-riemann-zeta-function}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-13}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

Define the infimal exponent $μ_{ζ}$ by $μ_{ζ} := in f {θ \geq 0 : ∣ ζ (1/2 + i t)∣ ≪_{ε} (1 + ∣ t ∣)^{θ + ε} for all ε > 0} .$

We define $C_{62 a} := μ_{ζ},$ the Lindelof (pointwise growth) exponent for $ζ (1/2 + i t)$ .

The Lindelof hypothesis is the conjecture that $∣ ζ (1/2 + i t)∣ ≪ t^{ε}$ for every $ε > 0$ , which is equivalent to $C_{62 a} = 0$ . <a href="#Har2019-lindelof">[Har2019-lindelof]</a>

Unconditionally, convexity gives $C_{62 a} \leq 1/4$ . <a href="#Har2019-convexity-1-4">[Har2019-convexity-1-4]</a> Hardy-Littlewood proved the bound $C_{62 a} \leq 1/6$ . <a href="#Har2019-hl-1-6">[Har2019-hl-1-6]</a> Bourgain proved the sharper bound $C_{62 a} \leq 13/84$ . <a href="#Bou2017-13-84">[Bou2017-13-84]</a>

The best established range currently is $0 \leq C_{62 a} \leq \frac{13}{84} .$

Known upper bounds

Bound	Reference	Comments
$1/4$	<a href="#Har2019">[Har2019]</a>	Convexity bound. <a href="#Har2019-convexity-1-4">[Har2019-convexity-1-4]</a>
$1/6$	<a href="#Har2019">[Har2019]</a>	Hardy-Littlewood bound. <a href="#Har2019-hl-1-6">[Har2019-hl-1-6]</a>
$13/84$	<a href="#Bou2017">[Bou2017]</a>	Bourgain's pointwise bound for $∣ ζ (1/2 + i t)∣$ . <a href="#Bou2017-13-84">[Bou2017-13-84]</a>

Known lower bounds

Bound	Reference	Comments
$0$		Trivial from the definition $C_{62 a} \geq 0$ .

Additional comments and links

Conjectural value. The Lindelof hypothesis predicts $C_{62 a} = 0$ . <a href="#Har2019-lindelof">[Har2019-lindelof]</a>

Wikipedia page on the Lindelof hypothesis

References

<a id="Bou2017"></a>[Bou2017] Bourgain, Jean. Decoupling, exponential sums and the Riemann zeta function. Journal of the American Mathematical Society 30 (2017), no. 1, 205-224. DOI: https://doi.org/10.1090/jams/860. arXiv PDF: https://arxiv.org/pdf/1408.5794.pdf. Google Scholar
<a id="Bou2017-13-84"></a>[Bou2017-13-84] loc: arXiv PDF p.1, Abstract quote: "In particular, this leads to an improved bound $∣ ζ (1/2 + i t)∣ ≪ t^{13/84 + ε}$ for the zeta function on the critical line."

<a id="Har2019"></a>[Har2019] Harper, Adam. La fonction zeta de Riemann dans les petits intervalles. Seminaire Bourbaki, Expose 1159 (2017/2018), Asterisque 414 (2019), 429-464. PDF: https://www.bourbaki.fr/TEXTES/Exp1159-Harper.pdf. Google Scholar
<a id="Har2019-convexity-1-4"></a>[Har2019-convexity-1-4] loc: PDF p.19, Section 2.4 quote: "General complex analysis arguments ('convexity') can prove a bound $∣ ζ (1/2 + i t)∣ ≪_{ε} t^{1/4 + ε}$ ."
<a id="Har2019-hl-1-6"></a>[Har2019-hl-1-6] loc: PDF p.19, Section 2.4 quote: "Long ago Hardy and Littlewood proved the bound $∣ ζ (1/2 + i t)∣ ≪_{ε} t^{1/6 + ε}$ ."
<a id="Har2019-lindelof"></a>[Har2019-lindelof] loc: PDF p.19, Section 2.4 quote: "The classical Lindelof Hypothesis ... conjectures that $∣ ζ (1/2 + i t)∣ ≪_{ε} t^{ε}$ for any $ε > 0$ and all large $t$ ."

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.