## Constants and Records of Computation

Last update: August 12 2010

Ten decimal places of are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope. Simon Newcomb (1835-1909)

I myself have yielded to the temptation of calculating quantities that interested me to a greater number of decimals than could ever be required, and can testify to the pleasure of admiring a long row of figures in a new result and realizing what a veritable triumph of algebra and arithmetic they represent. James Whitbread Lee Glaisher (1848-1928)

The following table illustrates the various difficulties encountered to compute some famous Mathematical Constants with as many digits as possible. The computation of many decimal places of those constants has involved for each of them a few and sometimes a lot of mathematicians.

It's of interest to notice that the methods used for this purpose are numerous and cover a large panel of classical algorithms: iterative algorithms, series expansion, binary splitting, FFT, numerical quadratures, sieve, Newton's iteration, eigenvalues, ... Of course all those constants have not attracted as much interest as the celebrated constant p and therefore it only gives an idea of the computational complexity of each of them. Observe that the memory required to store so many digits becomes huge for the last constants (and especially in Kanada's, Daisuke's, Bellard's or Yee-Kondo's latest impressive computation of the constant p) on a super PC.

Other interesting constants may be included in this table and the current records will be updated when necessary. To avoid mistakes a record must be established by two, as independent as possible, methods.

 First digits Computed digits Who - Year Brun's constant 1.902160582... 9 T. Nicely - 1999 & P. Sebah - 2002 Gauss-Kuzmin-Wirsing 0.30366300289873265... 468 K. Briggs - 2003 Artin's constant 0.37395581361920228... 1,000 G. Niklasch - 1999 Fransén-Robinson 2.80777024202851936... 1,025 P. Sebah - 2001 Twin prime constant 0.66016181584686957... 5,020 P. Sebah - 2001 Mertens' constant 0.26149721284764278... 8,010 P. Sebah - 2001 Landau-Ramanujan K 0.76422365358922066... 30,010 P. Sebah - 2002 Soldner-Ramanujan 1.45136923488338105... 75,500 P. Sebah - 2001 Khintchine's constant 2.68545200106530644... 110,000 X. Gourdon - 1998 G(1/4) 3.62560990822190831... 10,000,000,000 S. Kondo & S. Pagliarulo - 2010 G(1/3) 2.67893853470774763... 10,000,000,000 S. Kondo & S. Pagliarulo - 2009 Euler's constant g 0.57721566490153286... 29,844,489,545 R. Chan & A.J. Yee - 2009 Catalan's constant G 0.91596559417721901... 31,026,000,000 R. Chan & A.J. Yee - 2009 z(3) 1.20205690315959428... 31,026,000,000 R. Chan & A.J. Yee - 2009 log 2 0.69314718055994530... 31,026,000,000 R. Chan & A.J. Yee - 2009 Golden ratio f 1.61803398874989484... 1,000,000,000,000 A.J. Yee - 2010 e 2.71828182845904523... 1,000,000,000,000 S. Kondo & A.J. Yee - 2010 Ö2 1.41421356237309504... 1,000,000,000,000 S. Kondo & A.J. Yee - 2010 p 3.14159265358979323... 5,000,000,000,000 S. Kondo & A.J. Yee - 2010

The last constants of this table are fully detailed in our pages. The other one are less famous and are introduced in Steven Finch's encyclopedia: Mathematical Constants (Cambridge University Press). Here we only give of them a short description as well as some previous known high accuracy computations.

• Brun's constant B is defined as the sum of the inverse of the twin primes, that is primes p such as p + 2 is also a prime:

B = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ...

Brun showed in 1919 that this sum converges to a finite limit. In order to estimate this sum the only known algorithms are variant of classical sieve to compute all twin primes as far as possible. This constant is famous because T. Nicely found the famous Pentium bug during its computation... To achieve his  estimation,  he needed to find all twin primes up to about 3.1015 ! Note that the twin prime's constant is used in the estimation of this constant. The latest computation of this constant was achieved up to 1016 by P. Sebah in July 2002 and required a cluster of 16 computers working during two months.

Other computations:

• E.S. Selmer: 2 (1942)
• C.E. Fröberg: 3 (1961)
• D. Shanks & J.W. Wrench: 4 (1974)
• J. Bohman: 5 (1974)
• R.P. Brent: 6 (1975)
• P. Sebah: 8 (1999)
• The Gauss-Kuzmin-Wirsing constant is in fact negative and plays an important role in the behavior of the regular continued fraction expansion of a random  number. The process is detailed in Knuth's famous The Art of Computer Programming (Addison Wesley, vol. 2) and this constant can be computed as the eigenvalue of a suitable linear operator involving integer values of the Zeta function

Other computations:

• E. Wirsing: 20
• P. Flajolet & B. Vallée: 35 (1995)
• J. Hershberger: 40 (1997)
• P. Sebah: 100 (2000)
• K. Briggs: 385 (2003)
• Artin's constant and Twin prime's constant are infinite products involving only prime numbers and are important in number theory. Accurate computations are made by mean of a fast evaluations of the Zeta function at integer values.

Other computations:

• J.W. Wrench: 45 (both constants 1961)
• T.O. Silva: 500 (only Artin's constant)
• G. Niklasch: 1,000 (1999)
• The Fransén-Robinson constant is defined as a definite integral related to the gamma function G(x). The algorithms used are various numerical quadratures (Newton-Cotes, Gauss-Legendre and Clenshaw-Curtis for the current record).

Other computations:

• H.P. Robinson: 36 (1979?)
• A. Fransén & S. Wrigge: 80 (1984)
• W. A. Johnson: 300
• Mertens' constant M is similar as Euler's constant but this time only the primes are taken in account in the sum. Hence it is the limit of

1/2 + 1/3 + 1/5 + 1/7 + ... + 1/p - log log p

as p tends to infinity. To achieve this you need to evaluate the Zeta function to a great accuracy for successive integer values.

Other computations:

• G. Niklash: 1,000 (1999)
• P. Sebah: 4,050 (2000)
• Landau-Ramanujan's constant K occurs in the estimation of the number N(x) of positive integers less than x which can be expressed as the sum of two squares. We have the property that (Landau 1908)

Other computations:

• S. Ramanujan: 3
• P. Flajolet: 1,024 (1996)
• V. Adamchik, J. Golden & W. Gosper: 5,000
• D. Hare: 10,000 (1996)
• Soldner-Ramanujan's constant is the zero of the integral logarithm Li(x) function. It was computed using a fourth order Newton's iteration and a fast evaluation of the function Li(x).

Other computations

• S. Ramanujan: 4
• J. von Soldner: 9
• P. Sebah: 10,000 (1999)
• Khintchine's constant is the limit, as n tends to infinity, of the geometric mean of the first n partial quotients of the simple continued fraction for almost any real number x. It is a remarkable fact that this limit exists and is a constant independent from x for most numbers.

Other computations

• R.W. Gosper: 2,217 (1997?)
• D.H. Bailey, J.M. Borwein & R.E. Crandall: 7,350 (1997)
• The transcendental constants G(1/3) and G(1/4) were both computed using the AGM iteration. Very efficient algorithms were used to compute both constants.

Other computations:

• G.J. Fee & S. Plouffe: 250,000 (both constants in 1996)
• S. Spännare: 16,693,288 (2003)
• P. Sebah & M. Tommila: 51,097,000 (only G(1/4) in 2001)
• Catalan's constant G may be defined by means of the slowly alternating convergent series

G = 1/12 - 1/3 + 1/5- 1/7+ 1/92 - ...

Other computations:

• J.W.L. Glaisher: 20 (1877)
• J.W.L. Glaisher: 32 (1913)
• G.J. Fee: 20,000 (1990)
• G.J. Fee: 50,000 (1996)
• G.J. Fee & S. Plouffe: 100,000 (1996)
• T. Papanikolaou: 300,000 (1996)
• T. Papanikolaou: 1,500,000 (1996)
• P. Demichel: 3,379,957 (1997)
• X. Gourdon: 12,500,000 (1998)
• X. Gourdon & P. Sebah: 50,010,000 (2001)
• X. Gourdon & P. Sebah: 100,000,500 (2001)
• S. Kondo & S. Pagliarulo: 2,000,000,000 (2007)
• S. Kondo & S. Pagliarulo: 10,000,000,000 (2008)
• R. Chan & A.J. Yee: 15,510,000,000 (2009)
• The Golden Ratio f is defined by (1 + Ö5)/2.

Other computations:

• M. Maestlin: 7 (1597)
• M. Berg: 4,599 (1966)
• J. Shallit: 10,000 (1976)
• G.J. Fee & S. Plouffe: 10,000,000 (1996)
• X. Gourdon & P. Sebah: 1,500,000,000 (2000)
• X. Gourdon & P. Sebah: 3,141,000,000 (2002)
• A. Irlande: 5,000,000,000 (2007)
• S. Kondo & S. Pagliarulo: 10,000,000,000 (2007)
• A. Irlande: 10,000,000,000 (2007)
• X. Gourdon & P. Sebah: 16,380,340,000 (2008)
• A. Irlande: 17,000,000,000 (2008)
• X. Gourdon & P. Sebah: 31,415,927,000 (2008)

Please, let us know if the number of correct digits of any of those constants has been improved or if a previous computation should appear here.