## Abstract

In 2002, L.N. Trefethen of Oxford University challenged the scientific community by ten intriguing mathematical problems to be solved, numerically, to ten digit accuracy each (I was one of the successful contestants; in 2004, jointly with three others of them, I published a book—Bornemann et al.: The SIAM 100-Digit Challenge, SIAM, Philadelphia, 2004—on the manifold ways of solving those problems). In this paper, I collect some new and noteworthy insights and developments that have evolved around those problems in the last decade or so. In the course of my tales, I will touch mathematical topics as diverse as divergent series, Ramanujan summation, low-rank approximations of functions, hybrid numeric-symbolic computation, singular moduli, self-avoiding random walks, the Riemann prime counting function, and winding numbers of planar Brownian motion. As was already the intention of the book, I hope to encourage the reader to take a broad view of mathematics, since one lasting moral of Trefethen’s contest is that overspecialization will provide too narrow a view for one with a serious interest in computation.

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## Notes

- 1.
Trefethen got the idea for this type of course from Don Knuth at Stanford, who got it from Bob Floyd, who got it from George Forsythe, who got it from George Pólya [6, p. 6].

- 2.
The only other example known to me is the 1794

*Thesaurus logarithmorum completus*of Jurij Vega, which was advertised as a carefully corrected edition of the Vlacq table. Though heavily criticized by Gauss in 1851 [19, p. 259] for a lot of numbers being wrong in several units in the last place, it remained popular for accurate calculations until after the dawn of the computer age. - 3.
Part of the motivation was that, tongue-in-cheek, Trefethen had asked when announcing the winners of the contest: “If you add together our heroic numbers the result is \(\tau= 1.497258836\ldots\) . I wonder if anyone will ever compute the ten thousandth digit of this fundamental constant?”

- 4.
I will cheerfully award 273€ (and Trefethen has offered to double the pot to 546€) to the first individual or team that convinces me that they have succeeded in calculating these elusive \(10\,000\) digits and, thus, the ten thousandth digit of Trefethen’s constant \(\tau\).

- 5.
The issue of rounding vs. truncation muddied the actual scoring of the contest: within a few days after the initial posting of the results, the list of 100-point winners grew by two.

- 6.
Note that, at face value, \(\zeta (3)= 1.20205\,69032\) is simply wrong and \(\zeta(3) = 1.20205\, 69032\ldots\) would be rather misleading, even though the latter can be found frequently in the literature.

- 7.
In the scientific computing literature numerical experiments, often documented in form of plots, are mostly meant to be illustrations of some facts about the algorithm itself. On the other hand, more often than not, the user displays the results of an application run in form of plots of limited and undocumented accuracy, too.

- 8.
At the turn of the 19th century it was common to announce awards for finding errors in a mathematical table. For instance, Vega’s 1794

*Thesaurus logarithmorum completus*offered a ducat for each error, which is more than $100 worth in today’s money. In his 1851 review [19, p. 260], estimating that probably more than 5 670 errors are to be found in the log-trigonometric values, Gauss dryly remarked that, if such an award had still applied, it would had been cheaper to commission able people with a complete recalculation of the table. - 9.
One of the lesser known special functions, defined by the equation \(x = W(x) e^{W(x)}\) [14].

- 10.
There is a derivation of the divergent series (1) that provides much more context than the repeated integration by parts alluded to by Strang. Substituting \(t\) for \(-\log x\) gives

$$S = \int_{0}^{\infty}\cos\bigl(t e^{t}\bigr)\,dt $$and, therefore, (1) is obtained by formally setting \(z = i\), and taking real parts, in the following asymptotic expansion, which can be obtained by Laplace’s method [36, p. 125]:

$$\int_{0}^{\infty}e^{-z t e^{t}}\,dt \sim\sum _{n=1}^{\infty}\frac{(-1)^{n-1} n^{n-1}}{z^{n}}, $$as \(z\to\infty\) in a closed sector of the half plane \(|\arg z| < \pi /2\). This way, the relation to analytic continuation becomes clearly visible.

- 11.
Quite heroically back in 1928, Watson took pride in correct digits: “In default of any less laborious method, I computed this integral by the Euler–Maclaurin sum-formula.”

- 12.
- 13.
In this vein, I get the value of Ramanujan’s fifth series as the Laplace integral

$$1-1^{1}+2^{2}-3^{3}+4^{4}-5^{5}+ \cdots= \int_{0}^{\infty}\frac{e^{-x}}{1+W(x)}\,dx, $$which is easily transformed into Watson’s integral (4) by the substitution \(t=x/W(x)\).

- 14.
For this limit to hold, I have used the uniform bound \(|z W'(z)|\leq1\) for \(\operatorname {Re}z \geq0\), which easily follows from the Stieltjes integral representation [27, Eq. (8)]

$$W'(z) = \frac{1}{\pi} \int_{0}^{\pi}\frac{d\tau}{z + \tau\csc(\tau) e^{-\tau \cot\tau}}\quad\bigl(|\arg z|< \pi\bigr). $$ - 15.
For download at www.chebfun.org.

- 16.
The edition history of Ramanujan’s notebooks is quite convoluted, see [2, p. 5]. A lack of funds prevented the notebooks from being published with the

*Collected papers*in 1927. At this time, the originals were kept at the University of Madras, while Hardy was in the possession of handwritten copies. Watson and Bertram M. Wilson agreed in 1929 (just a year after Watson’s paper [52] on Ramanujan’s divergent series was written) to edit the notebooks, but Wilson died young in 1935 and Watson’s interest waned in the late 1930’s. In 1957, the Tata Institute published a photostat edition of the notebooks in two volumes. Bruce Berndt started his monumental five volume edition and commentary in the 1980’s, the first volume appeared in 1985, the last one in 1998. The five volume edition and commentary of Ramanujan’s*Lost Notebook*, by George E. Andrews and Berndt, is still work in progress, the first volume appeared in 2005, the fourth one in 2013. - 17.
I use the current definition of the Bernoulli numbers \(B_{k}\), namely

$$\frac{x}{e^{x}-1} = \sum_{n=0}^{\infty}\frac{B_{n}}{n!} x^{n}\quad \bigl(|x|< 2\pi\bigr). $$ - 18.
For more on that issue, and the recent work of Bernard Candelpergher and Éric Delabaere on Ramanujan’s method, see [12].

- 19.
If \(f\) is holomorphic, and \(O(|z|^{s})\), where \(s>0\), in the half plane \(\operatorname {Re}z \geq \delta\), where \(\delta< 1\), then Hardy [24, Thm. 246] proved that the Borel sum of this Euler–Maclaurin series has the value \(c_{a}\).

- 20.
I am quite sure that Ramanujan would have supplied more digits if his method had enabled him to do so: in his notebook he gives a table of the values of \(\zeta(k)\), for \(k=2,\ldots,10\), with a precision of eleven digits, all calculated, of course, by the Euler–Maclaurin formula.

- 21.
In [11], you will find that the summation formulas of Euler–Maclaurin, of Abel–Plana, and of Poisson are all equivalent, in the sense that each is a corollary of any of the others.

- 22.
Because of the quadratic term it is easy to see that the minimum is taken in the unit circle. There are 2720 critical points of \(f\) inside the square \([-1,1]\times[-1,1]\), see [6, §4.4].

- 23.
Chebfun has grown to considerable sophistication with quite a number of contributors, the most recent version 5.3 can be downloaded at www.chebfun.org.

- 24.
Employing the command fmincon of Matlab’s optimization toolbox.

- 25.
You are invited to solve the

*three-dimensional*problem in [6, §4.7] along these lines. - 26.
I discussed this in the 2006 German edition [7, §7.3] of the Challenge Book. The 2004 English original confined itself to the diagonally preconditioned CG method.

- 27.
- 28.
See www.numerikstreifzug.de for the \(97\,389\)-digits numerator and denominator.

- 29.
Here I have used \(|x_{1}|< 1\), which is already known from the ten-digit numerical values.

- 30.
Since we are interested in the first solution component \(x_{1}\) only, it suffices to store just the first components of the integer vectors \(\xi_{k}\) (and, in the same vein, to calculate just the first component of \(\xi\))—in Problem 7, this cuts down the storage requirement by a decisive factor of \(20\,000\).

- 31.
You can find the hybrid Matlab/

*Mathematica*code at www.numerikstreifzug.de. - 32.
His study led Kronecker to his conjecture about abelian extensions of imaginary quadratic fields, his famous “Jugendtraum”, which was solved by one of the pearls of 20th century pure mathematics, class field theory.

- 33.
This probability is quite small, after all: \(p=3.83758\,79792\,51226\ldots\cdot10^{-7}\). In Oxford’s

*Balliol College Annual Report 2005*Strang reported on some communication initiated by his review of the Challenge Book in*Science*:An email just came from Hawaii, protesting that the probability should not be so small. Comparing the side lengths, 1 chance in 10 or 1 in 100 seemed more reasonable than 1 in \(2\,500\, 000\). It’s just very hard to reach the narrow end! My best suggestion was to draw the rectangle in the driveway, and make a blindfold turn before each small step.

- 34.
There, he disclosed also that he was part of a team that entered the contest, but failed to qualify for an award [8, p. 44]: “We took Nick at his word and turned in 85 digits! We thought that would be a good enough entry and returned to other activities.”

- 35.
3Ms =

*Mathematica*, Maple, Matlab. - 36.
A variant of this derivation, using commands that are deprecated in

*Mathematica*now, is included in the German edition of the Challenge Book [7, §10.7.1]. - 37.
The command RootApproximant uses the famous LLL lattice reduction algorithm [30].

- 38.
Clisby is an expert on the efficient sampling of self-avoiding walks [13]. If you have ever played the computer game

*Snake*, you will appreciate the combinatorial difficulty of generating a self-avoiding walk of, say, \(100\,000\,000\) steps on the square lattice. Clisby created a striking video, zooming in and out of one such sample: www.youtube.com/watch?v=WPUH8Rs-oig. - 39.
Published in a special issue of the

*Journal on Engineering Mathematics*, dedicated to the memory of the eminent Milton van Dyke (1922–2010), master of perturbation analysis in aeronautics and creator of the book*An Album of Fluid Motion*. - 40.
For an approachable discussion of \(\mathrm{SLE}_{\kappa}\), see [35].

- 41.
As of today, 15 out of the 22 new problems have been solved: www.numerikstreifzug.de. Actually, two of them have initiated some published research, see [50] and [43].

- 42.
The first one, conjectured by Euler in 1748 and proved by von Mangoldt in 1897, was shown in 1911, by Landau, to be “elementarily” equivalent to the prime number theorem. The second one, conjectured by Möbius in 1832, was proved by Landau in 1899. The third one follows from an application of Möbius inversion to the Dirichlet series of \(\zeta(s)\).

- 43.
Wagon asked the late Richard Crandall “if he saw any value in knowing the largest zero of \(R(x)\)—and he said NO (though he liked the numerical problem)” (email of Aug. 19, 2005).

- 44.
In my implementation I used the table of the 100 smallest nontrivial zeros \(\rho\) to 1000 digit accuracy that Andrew Odlyzko maintains at www.dtc.umn.edu/~Odlyzko/zeta_tables.

- 45.
Personal communication to S. Wagon (email from Aug. 15, 2008).

- 46.
The sequence of its digits has made it into N.J.A. Sloane’s

*On-Line Encyclopedia of Integer Sequences*: oeis.org/A143531. - 47.
Once you have found it, you can verify it by inserting it into the constitutive equations.

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## Acknowledgements

I thank Martin Hanke-Bourgeois for having solicited this paper; it was presented first at the conference “New Directions in Numerical Computing: In celebration Nick Trefethen’s 60th birthday”, Oxford, August 27, 2015. Nick Trefethen has kindly ironed out some of my worst English language peculiarities.

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*Dedicated to L.N. Trefethen at the occasion of his 60th birthday.*

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Bornemann, F. The SIAM 100-Digit Challenge: A Decade Later.
*Jahresber. Dtsch. Math. Ver.* **118, **87–123 (2016). https://doi.org/10.1365/s13291-016-0137-2

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### Keywords

- Divergent series
- Numeric-symbolic methods
- Low-rank approximation
- Singular moduli
- Self-avoiding random walk
- Riemann \(R\) function