Lindelof (pointwise growth) exponent for the Riemann zeta function
Description of constant
Define the infimal exponent $\mu_\zeta$ by \(\mu_\zeta\ :=\ \inf\Bigl\{\theta\ge 0:\ \lvert\zeta(1/2+it)\rvert\ll_{\varepsilon}(1+\lvert t\rvert)^{\theta+\varepsilon}\ \text{for all }\varepsilon>0\Bigr\}.\)
We define \(C_{62a}\ :=\ \mu_\zeta,\) the Lindelof (pointwise growth) exponent for $\zeta(1/2+it)$.
The Lindelof hypothesis is the conjecture that $\lvert\zeta(1/2+it)\rvert\ll t^{\varepsilon}$ for every $\varepsilon>0$, which is equivalent to $C_{62a}=0$. [Har2019-lindelof]
Unconditionally, convexity gives $C_{62a}\le 1/4$. [Har2019-convexity-1-4] Hardy-Littlewood proved the bound $C_{62a}\le 1/6$. [Har2019-hl-1-6] Bourgain proved the sharper bound $C_{62a}\le 13/84$. [Bou2017-13-84]
The best established range currently is \(0\ \le\ C_{62a}\ \le\ \frac{13}{84}.\)
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1/4$ | [Har2019] | Convexity bound. [Har2019-convexity-1-4] |
| $1/6$ | [Har2019] | Hardy-Littlewood bound. [Har2019-hl-1-6] |
| $13/84$ | [Bou2017] | Bourgain’s pointwise bound for $\lvert\zeta(1/2+it)\rvert$. [Bou2017-13-84] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0$ | Trivial from the definition $C_{62a}\ge 0$. |
Additional comments and links
-
Conjectural value. The Lindelof hypothesis predicts $C_{62a}=0$. [Har2019-lindelof]
References
- [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
- [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 $\lvert\zeta(1/2+it)\rvert\ll_{\varepsilon} t^{1/4+\varepsilon}$.”
- [Har2019-hl-1-6] loc: PDF p.19, Section 2.4 quote: “Long ago Hardy and Littlewood proved the bound $\lvert\zeta(1/2+it)\rvert\ll_{\varepsilon} t^{1/6+\varepsilon}$.”
- [Har2019-lindelof] loc: PDF p.19, Section 2.4 quote: “The classical Lindelof Hypothesis … conjectures that $\lvert\zeta(1/2+it)\rvert\ll_{\varepsilon} t^{\varepsilon}$ for any $\varepsilon>0$ and all large $t$.”
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.