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Lindelof (pointwise growth) exponent for the Riemann zeta function (C62aC_{62a})

Lindelof (pointwise growth) exponent for the Riemann zeta function: determine C62aC_{62a} or improve its bounds; the repository records 0 (lower) and 1384\frac{13}{84} (upper).

MSC 11M06 · ζ(s)\zeta (s) and L(s,χ)L(s, \chi)

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Define the infimal exponent μζ\mu_\zeta by

μζ := inf⁡{θ≥0: ∣ζ(1/2+it)∣≪ε(1+∣t∣)θ+ε for all ε>0}.\mu_\zeta\ :=\ \inf\Bigl\{\theta\ge 0:\ \lvert\zeta(1/2+it)\rvert\ll_{\varepsilon}(1+\lvert t\rvert)^{\theta+\varepsilon}\ \text{for all }\varepsilon>0\Bigr\}.

We define

C62a := μζ,C_{62a}\ :=\ \mu_\zeta,

the Lindelof (pointwise growth) exponent for ζ(1/2+it)\zeta(1/2+it).

The Lindelof hypothesis is the conjecture that ∣ζ(1/2+it)∣≪tε\lvert\zeta(1/2+it)\rvert\ll t^{\varepsilon} for every ε>0\varepsilon>0, which is equivalent to C62a=0C_{62a}=0. [Har2019-lindelof]

Unconditionally, convexity gives C62a≤1/4C_{62a}\le 1/4. [Har2019-convexity-1-4] Hardy-Littlewood proved the bound C62a≤1/6C_{62a}\le 1/6. [Har2019-hl-1-6] Bourgain proved the sharper bound C62a≤13/84C_{62a}\le 13/84. [Bou2017-13-84]

The best established range currently is

0 ≤ C62a ≤ 1384.0\ \le\ C_{62a}\ \le\ \frac{13}{84}.

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Background

References (from the repository page)

  • [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

  • [Bou2017-13-84] loc: arXiv PDF p.1, Abstract quote: "In particular, this leads to an improved bound ∣ζ(1/2+it)∣≪t13/84+ε\lvert\zeta(1/2+it)\rvert\ll t^{13/84+\varepsilon} for the zeta function on the critical line."

  • [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

  • [Har2019-convexity-1-4] loc: PDF p.19, Section 2.4 quote: "General complex analysis arguments ('convexity') can prove a bound ∣ζ(1/2+it)∣≪εt1/4+ε\lvert\zeta(1/2+it)\rvert\ll_{\varepsilon} t^{1/4+\varepsilon}."

  • [Har2019-hl-1-6] loc: PDF p.19, Section 2.4 quote: "Long ago Hardy and Littlewood proved the bound ∣ζ(1/2+it)∣≪εt1/6+ε\lvert\zeta(1/2+it)\rvert\ll_{\varepsilon} t^{1/6+\varepsilon}."

  • [Har2019-lindelof] loc: PDF p.19, Section 2.4 quote: "The classical Lindelof Hypothesis ... conjectures that ∣ζ(1/2+it)∣≪εtε\lvert\zeta(1/2+it)\rvert\ll_{\varepsilon} t^{\varepsilon} for any ε>0\varepsilon>0 and all large tt."

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0; best known upper bound 1384\frac{13}{84}. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
1/41/4 [Har2019] Convexity bound. [Har2019-convexity-1-4]
1/61/6 [Har2019] Hardy-Littlewood bound. [Har2019-hl-1-6]
13/8413/84 [Bou2017] Bourgain's pointwise bound for ∣ζ(1/2+it)∣\lvert\zeta(1/2+it)\rvert. [Bou2017-13-84]

Known lower bounds

Bound Reference Comments
00 Trivial from the definition C62a≥0C_{62a}\ge 0.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Lindelof (pointwise growth) exponent for the Riemann zeta function ($C_{62a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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