One of the algorithms described here, here, and here permits the numerical evaluation of of $\zeta(\sigma + iT)$ in $T^{1/3+o(1)}$ time. This algorithm is useful in studying the zeta function in small neighbourhoods where extreme behavior was found likely. For example, there is a heuristic method due to Odlyzko that quickly finds values of $T$ where $|\zeta(1/2 + iT)|$ is probably large. On obtaining such a candidate value of $T$, one can compute zeta in a small neighbourhood of $T$ using the $T^{1/3}$ algorithm. A typical such computation consists of numerically evaluating $\zeta(1/2 + iT + it)$ on a dense enough grid for $-20 < t < 20$ say.
In contrast to the $T^{1/3}$ method, the amortized-complexity algorithm (see here and here) is more suited for mass computations of zeta; e.g. to test predictions from random matrix theory. So unlike the $T^{1/3}$ method, the amortized algorithm is not designed for studying zeta in "targeted neighbourhoods" (e.g. in neighbourhoods where extreme behavior is probable). This is because the amortized algorithm requires a precomputation costing about $T^{1/2}$ time, regardless of how many points are to be computed.
It is not clear what the smallest complexity exponent (up to an epsilon) that an algorithm for computing $\zeta(1/2+it)$ at a single point can be (note that on the Lindelöf hypothesis, zeta can be approximated by arbitrarily short dirichlet polynomial to the right of the critical line). It seems that obtaining a $T^{1/4}$ algorithm will be challenging as it requires some new ideas. Currently, the fastest available algorithm for evaluating zeta at a single point (to within $t^{-k}$ for any fixed $k$) has an asymptotic running time of about $T^{4/13+o(1)}$; see here. Notice $4/13$ is approx. $0.307$.
It is worth noting that in improving the complexity exponent of such "targeted" analytic algorithms, one often encounters what looks like a "natural barrier", a complexity exponent that seems hard to break. For zeta, the exponent 1/3 is such a barrier. In this case, the $T^{4/13}$ algorithm shows that this barrier can be broken by a noticeable margin.
GitHub repository with $T^{1/3}$ algorithm code to compute zeta main sum.
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| $T$ | sample data for $Z(t)$: using a few hundred zeros near each height $T$ | |||
|---|---|---|---|---|
| 88837796029624663862630219091085 (1e32) | zero ordinates | S(t) values | Z(t) values | Plot |
| 70391066310491324308791969554433 (1e32) | __ | __ | __ | __ |
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(1) The LLL algorithm is used to find $y_0$ such that $|\prod_{p \le P} (1 - p^{-1/2-iy_0})^{-1}|$ is large. Usually $P$ was in the range 200--1000.
(2) The main sum of zeta is computed on a grid of points centered around $y_0$. Specifically, one computes $\sum_{n \le N} n^{-1/2-iy}$, where $N = \lfloor \sqrt{y/2 \pi}\rfloor$, on the grid $y = y_0 + 0.04 r$, $r = -500,\ldots, 500$.
(3) A band-limited interpolation method is used to recover values of zeta between grid points. The maximum is then located using a standard bisection algorithm.
The largest value of $|\zeta(1/2+it)|$ found this way was at $t$ = 3.9246764589894309155251169284104050622201, where $|\zeta(1/2+it)|=16244.865266...$. There is usually a wide zero gap around such large values of zeta. So $|S(t)|$ tends to get large there also. Among the large values of $|S(t)|$ found this way, the largest was at $t$ = 7.75730499036786141715021305363868444e27, where $S(t) = 3.345544311$.| $T$ | $t_{max}$ | $Z(T+t_{max})$ | $\mu_T(t_{max})$ | $u_{max}$ | $S(T+u_{max})$ | Plot |
| 39246764589894309155251169284084 (1e32) | 20.050622201 | 16244.8652669745 | 0.13328 | 19.7774677919 | 3.1694562060 | __ |
| 70391066310491324308791969554433 (1e32) | 20.2490986212 | -14055.8928570577 | 0.13024 | 20.4878399509 | -2.8592375097 | __ |
| 552166410009931288886808632326 (1e30) | 20.5052428685 | -13558.8331415378 | 0.13894 | 20.2293397550 | 2.9427260298 | __ |
| 35575860004214706249227248805957 (1e32) | 20.2412877269 | 13338.6875635420 | 0.13074 | 19.9763320821 | 3.2722532684 | __ |
| 6632378187823588974002457910686 (1e31) | 20.5963776248 | 12021.0940441987 | 0.13237 | 20.1556463240 | 2.8123873969 | __ |
| 698156288971519916135942940440 (1e30) | 20.333708866 | 11196.7919996115 | 0.13568 | 20.6087218809 | -2.9817249137 | __ |
| 289286076719325307718380549030 (1e29) | 20.2563707347 | 10916.1145229406 | 0.13706 | 19.7935302525 | 2.7765205945 | __ |
| 50054757231073962115880454671597 (1e32) | 20.4008687276 | -10622.1763650751 | 0.12701 | 20.8419170712 | -3.1826088138 | __ |
| 803625728592344363123814218758 (1e30) | 20.1993772725 | 10282.6496798924 | 0.13416 | 19.8386298158 | 2.9203959700 | __ |
| 690422639823936254540302269422 (1e30) | 20.4854435356 | 10268.7134126745 | 0.13444 | 20.0301671566 | 2.8040305852 | __ |
| 1907915287180786223131860607177 (1e30) | 20.5463535959 | 10251.5994011526 | 0.13245 | 20.2175641861 | 2.6168017337 | __ |
| 9832284408046499500622869540111 (1e31) | 20.7445640206 | -10138.5908684884 | 0.12926 | 21.1846414727 | -2.8818915011 | __ |
| 45890014847929271884961558864608 (1e32) | 20.1935624464 | 9978.02802184827 | 0.12631 | 19.7458906350 | 2.9634938648 | __ |
| 505734497867330197269203385120 (1e30) | 19.959637846 | 9763.37714228913 | 0.13431 | 20.4093647441 | -2.7784093185 | __ |
| 6083028695276545807063248346835 (1e31) | 20.9062681249 | 9739.33552572717 | 0.12956 | 21.2656017105 | -2.9926739520 | __ |
| 12120513446447725601585991067 (1e28) | 20.703844345 | 9693.82621946527 | 0.14195 | 20.4430730253 | 2.6148456269 | __ |
| 85052987797410463487724040101 (1e29) | 20.4017460411 | -9682.00496645840 | 0.13778 | 20.6797104905 | -2.9682163381 | __ |
| 46709141854660972368505489012 (1e29) | 20.7415908851 | 9589.21053972612 | 0.13889 | 21.0256762786 | -3.0349508458 | __ |
| 31774695316763918183637654345 (1e29) | 20.1151428181 | -9563.63957786792 | 0.13966 | 19.8066017585 | 3.2573734302 | __ |
| 840254377093000239632447060594 (1e30) | 20.4741455786 | 9474.56854936430 | 0.13289 | 20.2263459701 | 2.8477331190 | __ |
| 49520025290946288154611183520272 (1e32) | 20.9565206475 | 9316.94230380649 | 0.12523 | 20.5511155739 | 2.7862403553 | __ |
| 5534453662630304774681440886 (1e28) | 20.4862954317 | -9286.32881908699 | 0.14302 | 20.2026245830 | 2.8006627807 | __ |
| 2409972808744819410276834556260 (1e30) | 20.21875855 | -9269.15033230645 | 0.13057 | 20.6353913402 | -2.5871595918 | __ |
| 151430114521478793791897636280 (1e29) | 19.9404440337 | 9171.08674409048 | 0.13579 | 20.4053954126 | -2.9332504642 | __ |
| 546577562321057124801498516799 (1e30) | 20.1780843603 | 9122.70255584184 | 0.13317 | 20.4609831729 | -3.2748581152 | __ |
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| 1500000000000000000000000005000 (1e30) | 35.9862633272 | -56.4829980035973 | 0.05806 | 14.6319559956 | 1.9144065514 | __ |
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| 1500000000000000000000000008000 (1e30) | 17.8584170099 | 49.1144369522171 | 0.05604 | 31.4068662662 | -1.8608778687 | __ |
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| 1057983951339984806752281456 (1e27) | 35.1285243788 | -44.1574097870203 | 0.06087 | 14.1131856794 | 1.8657951191 | __ |
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| 10191135223869807023206505960 (1e28) | 22.5057661642 | -34.9445393164914 | 0.05510 | 17.1673599730 | 2.0843971847 | __ |