Lindelof (pointwise growth) exponent for the Riemann zeta function

Description of constant

Define the infimal exponent ${\mu }_{\zeta }$ by $${\mu }_{\zeta }:=\inf \{\theta \ge 0:|\zeta (1/2+it)|{\ll }_{\epsilon }(1+|t|{)}^{\theta +\epsilon }\text{for all }\epsilon >0\}.$$

We define $$C_{62a}:={\mu }_{\zeta },$$ the Lindelof (pointwise growth) exponent for $\zeta (1/2+it)$.

The Lindelof hypothesis is the conjecture that $|\zeta (1/2+it)|\ll t^{\epsilon }$ for every $\epsilon >0$, which is equivalent to $C_{62a}=0$. <a href="#Har2019-lindelof">[Har2019-lindelof]</a>

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

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

Known upper bounds

Known lower bounds

Additional comments and links

• Conjectural value. The Lindelof hypothesis predicts $C_{62a}=0$. <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 $|\zeta (1/2+it)|\ll t^{13/84+\epsilon }$ 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 $|\zeta (1/2+it)|{\ll }_{\epsilon }{t}^{1/4+\epsilon }$."
• <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 $|\zeta (1/2+it)|{\ll }_{\epsilon }{t}^{1/6+\epsilon }$."
• <a id="Har2019-lindelof"></a>[Har2019-lindelof] loc: PDF p.19, Section 2.4 quote: "The classical Lindelof Hypothesis ... conjectures that $|\zeta (1/2+it)|{\ll }_{\epsilon }{t}^{\epsilon }$ for any $\epsilon >0$ 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 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.