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on Lindelof (pointwise growth) exponent for the Riemann zeta function ($C_{62a}$)

The statement

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}.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.