# Apéry's constant $\zeta(3)$ fastest convergent series

UPDATE Feb.02.2024

The series below, Eq.(3) for computing and Eq.(2) for verifying, were applied by Andrew Sun on Dec.22.2023 to get over $$2\cdot10^{12}$$ decimal digits and break the number of currently known decimals of Apery's Constant. The whole process took about 12 days as it is reported in [1], [2] and [3].

It is worth to point out that these proven fast ζ(3) formulae Eqs.(1,2 & 3) are un-published and they have been only placed here in MO.

By means of a generalized search of hypergeometric-type series based on Dougall's sum and applying the Wilf-Zeilberger method as implemented by Jesús Guillera to prove some classical constants relationships, three proven series are found for Apéry's constant $$\zeta(3)$$. These series show the lowest known computing costs to apply the binary splitting method using A. Yee's y-cruncher software, a high performance platform to compute a huge number of digits of some classical constants, having set several records in this regard. These series are currently the fastest known to calculate an arbitrary number of digits for this constant.

I am not sure if any of these series has been published, so the question is simple.

Are these series known?

The first one is $$\begin{equation*}\zeta(3) = \frac{1}{12}\sum_{n=1}^{\infty}\frac{P(n)}{n^5(2n-1)(3n-1)(3n-2)(4n-1)(4n-3)\binom{3n}{n}^2\binom{6n}{3n}^2\binom{7n}{3n}}\tag{1}\label{1} \end{equation*}$$ with $$P(n) = 376698240\,n^7 - 935880048\,n^6 + 938536090\,n^5 - 491213175\,n^4 + 144859115\,n^3 - 24133842\,n^2 + 2114280\,n - 75600$$ It has convergence rate $$\rho = 1/22235661$$ giving about $$7.35$$ decimal digits per term. The computing cost $$C_s$$ is measured through $$C_s = - \frac{4\,d}{\log\rho}$$ where $$d$$ is the denominator's polynomial degree. In this case $$d = 10$$. This gives $$C_s=2.3645...$$.

The second series has $$C_s = 2.3070...$$ with $$d=14$$ and $$\rho=1/34828517376$$ giving about $$10.54$$ decimal digits per term. It is $$\begin{equation*}\zeta(3) = \frac{1}{24}\sum_{n=1}^{\infty}\frac{P(n)}{R(n)\cdot\binom{3n}{n}\binom{6n}{3n}\binom{8n}{4n}\binom{9n}{3n}\binom{10n}{5n}}\tag{2}\label{2} \end{equation*}$$ where $$P(n)= 250765325100000\,n^{11} - 1087318449630000\,n^{10} + 2067749814046250\,n^9 - 2269551612681475\,n^8 + 1592180015776565\,n^7 - 746938801646725\,n^6 + 238210943593421\,n^5 - 51452348050672\,n^4 + 7352050259484\,n^3 - 660416507568\,n^2 + 33552610560\,n - 731566080$$ and $$R(n)=n^5\,(2n-1)\,(3n-1)\,(3n-2)\,(4n-1)\,(4n-3)\,(5n-1)\,(5n-2)\,(5n-3)\,(5n-4)$$

The 3rd series has $$C_s=2.0514...$$ with $$d=14$$ and $$\rho=-1/717445350000$$ giving about $$11.86$$ decimal digits per term. It can be expressed as $$\begin{equation*}\zeta(3) = \frac{1}{48}\sum_{n=1}^{\infty}\frac{(-1)^{(n-1)}\,P(n)}{R(n)\cdot\binom{5n}{n}\binom{5n}{2n}\binom{9n}{4n}\binom{10n}{5n}\binom{12n}{6n}}\tag{3}\label{3} \end{equation*}$$ where $$P(n)= 1565994397644288\,n^{11} - 6719460725627136\,n^{10} + 12632254526031264\,n^9 - 13684352515879536\,n^8 + 9451223531851808\,n^7 - 4348596587040104\,n^6 + 1352700034136826\,n^5 - 282805786014979\,n^4 + 38721705264979\,n^3 - 3292502315430\,n^2 + 156286859400\,n - 3143448000$$ and $$R(n)=n^5\,(2n-1)^3\,(3n-1)\,(3n-2)\,(4n-1)\,(4n-3)\,(6n-1)\,(6n-5)$$

A non-native implementation of such algorithms has been done for y-cruncher software preparing custom input files for each formula. I tested these series under this special platform against some classical formulas that has been used to break the $$\zeta(3)$$ digits number record. In particular the Amdeberhan-Zeilberger series AZ (1997) $$C_s=2.8854...$$ and Wedeniwski's series Wed (1998) $$C_s=2.7554...$$.

Testings were done on an Intel Core Kaby Lake i5-7300U 3.5 Ghz 32 Gb RAM standard laptop with Windows 11 Pro 64bit OS. The following Table shows timings in seconds to get 50.000.000 decimal digits of Apéry constant. $$Arch$$ refers to the word size of the formula's input integer polynomial coefficients. $$N$$ indicates if the series summand has $$N$$ terms grouped from the basic formula. Timings are the average of 5 runs.

\begin{align*} \begin{array}{|c|c|c|c|c|c|c|} \hline {Series} & {Time\,s} & {Arch.} & N & \frac{dec.\ digits}{term} & C_s & d \\ \hline \mathrm{Eq.(3)} & 23.361 & 64-bit & 1 & 11.86 & 2.0514 & 14\\ \hline \mathrm{Eq.(2)} & 26.387 & 64-bit & 1 & 10.54 & 2.3070 & 14\\ \hline \mathrm{Eq.(1)} & 27.465 & 32-bit & 1 & 7.35 & 2.3645 & 10 \\ \hline \mathrm{Wed-II} & 33.095 & 64-bit & 2 & 10.08 & 2.7554 & 16\\ \hline \mathrm{AZ-III} & 34.697 & 64-bit & 3 & 9.03 & 2.8854 & 15\\ \hline \mathrm{Wed-I} & 34.893 & 32-bit & 1 & 5.04 & 2.7554 & 8\\ \hline \mathrm{AZ-I} & 37.603 & 32-bit & 1 & 3.01 & 2.8854 & 5\\ \hline \end{array} \end{align*}

Considering this limited scope of testings, it is observed that, under the y-cruncher's generic algorithm custom implementation, these -I guess- new series provide a significative increment of computing speed for Apéry's $$\zeta(3)$$. Eqs.$$(1), (2)$$ and $$(3)$$ are about 21%, 24% and 33% faster than classical Wedeniwski's formula respectively.

Finally, the proof of these series is obtained by means of WZ certificates (too long to put them here) through the code in Guillera with input patterns $$entry = 2+7\,n,1+3\,n,1+2\,n,1+1\,n,1+1\,n$$ $$entry = 2+10\,n,1+4\,n,1+3\,n,1+2\,n,1+1\,n$$ $$entry = 2+11\,n,1+4\,n,1+3\,n,1+2\,n,1+1\,n$$ for Eqs.$$(1), (2)$$ and $$(3)$$ respectively.

• Interesting idea that of adding the computation of the cost of binary splitting to my program Commented Sep 21, 2023 at 6:42
• What do your results imply about the irrationality measure of $\zeta(3)$? Commented Sep 21, 2023 at 12:47
• Muchas gracias a Ud. doctor Guillera, for providing the references without which it would have been impossible to discover these formulas. In particular, for making public the code to find and prove new series, which I could modify to automatically calculate the binary splitting computational costs $C_s$ Commented Sep 21, 2023 at 14:00
• Timothy Chow, this is a very difficult question. I have not (I guess nobody currently has) an answer. As it can be seen, WZ resulting formulas have a (very) high weight which makes them very difficult to theoretically work with. Commented Sep 21, 2023 at 14:29

• Thank you for your answer. The key is "those that have a fast convergence and small polynomials". This means to minimize the computing cost $C_s=\frac{-4d}{\log\rho}$. I have not worked with integrals. Just looking for Wilf-Zeilberger pairs from Dougall's well poised series where you have 5 parameters to work with, it is possible to find such fast formulae. Most of them come from a selected set of parameters. I think I have exhausted the searching of these series for $\zeta(3)$ with polynomials having 32-bit word size coefficients. Commented Sep 20, 2023 at 3:52
• Timothy, note that "fast" in my note must not be understood in the classical sense of "how many digits by term you get" for a given series, in other words it is not (only) related to the convergence rate $\rho$. The measurement of how fast a series is, is related to a concept of algorithmic exertion economy, which traduces in computing time. For computing certain hypergeometric forms with binary splitting algorithms we need first low degree polynomials and second a good convergence rate. The combination of both elements yields a fast series. Commented Sep 21, 2023 at 13:44