## Fast methods to compute the Riemann zeta function

• Preparation of this material is partially supported by the National Science Foundation under agreement No. DMS-1406190.
• 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.
• ### Some results and remarks:

• I implemented the $T^{1/3}$ algorithm with Jonathan Bober. We used it to compute zeta in neighbourhoods of about 200 large values, as well as some other random points. (See here for some graphs by Bober.) The largest $T$ where the algorithm was used in these computations was near $10^{32}$. It is possible to use the algorithm to compute a few hundred zeros around $T = 10^{36}$, say, in a reasonable amount of time.
• Let $Z(t)$ be the rotated zeta function. We have $|Z(t)| = |\zeta(1/2+it)|$, and $|Z'(t_j)| = |\zeta'(1/2+it_j)|$ where $t_j$ is a real zero ordinate. The following files contains data for $Z(t)$ obtained using the $T^{1/3}$, in case it is of interest. The values of $Z(t)$ in these files may not be more accurate to more that $5 \times 10^{-7}$. The zero ordinates may not be more accurate to more than $5\times 10^{-8}$. Turing's method was used to verify that the correct number of zeros was found in a certain subinterval (see the end of each zero ordinates file for detail). I often calculate $S(t)$ beyond this subinterval even though the calculation is completely justified within this subinterval only. The location of the max/min of $S(t)$ (i.e. $u_{max}^{\pm}$ below) may not be more accurate to more than $10^{-6}$.
 zero ordinates file format $T$ $n$ $z_1$ $z_2$ $\vdots$ $z_n$ Turing's method $Z(T + z_j) = 0$.
 $S(t)$ values file format $T$ $u^+_{max} \qquad S(T+u^+_{max})$ $u^-_{max} \qquad S(T+u^-_{max})$ $u_1 \qquad\,\,\,\,\, S(T+u_1)$ $\vdots$ $u_m \qquad\,\,\,\,\, S(T+u_m)$ $u_{j+1} - u_j = 0.01$.$\max_{t\in [u_1,u_m]} \pm S(T+t) =S(T+u^{\pm}_{max})$.
 $Z(t)$ values file format $T$ $w_{max} \quad\,\, Z(T+w_{max})$ $w_1 \qquad Z(T+w_1)$ $w_2 \qquad Z(T+w_2)$ $\vdots$ $w_r \qquad Z(T+w_r)$ $w_{j+1}-w_j = 0.01$. $\max_{t\in [w_1,w_r]} |Z(T+t)| = |Z(T+w_{max})|$.
Table of data for $Z(t)$ at various heights $T$ computed using the $T^{1/3}$ algorithm
$T$ 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
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• The following method (due to Odlyzko) was used to search for large values of $|\zeta(1/2+it)|$.

(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 $S(t) = 3.345544311$.
• Below are tables containing data about extreme values of $|\zeta(1/2+it)|$ and $S(t)$. We define $$\mu_T(x) = \frac{\log |Z(T+x)|}{\log (T+x)}.$$  $T$ $t_{max}$ $Z(T+t_{max})$ $\mu_T(t_{max})$ $u_{max}$ $S(T+u_{max})$ 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 __ 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