Approximate expressions for mathematical constants from PSLQ ...

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Aug 25, 2012 - NT] 25 Aug 2012. Approximate ... Another nice simple approximation is ζ(3) ≈ 4. √ γ + 71. 47. ,. (2) ..... II (Paris, 1992); Prog. Math., Birkhauser,.

arXiv:0910.2684v4 [math.NT] 25 Aug 2012

Approximate expressions for mathematical constants from PSLQ algorithm: a simple approach and a case study F. M. S. Lima Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia-DF, Brazil

Abstract In this note, I present a simple PSLQ code for finding null linear combinations, with the best rational coefficients, of mathematical constants, within some prescribed precision. As an P example, I explore approximate expressions for the Ap´ery’s constant ζ(3) = n≥1 1/n3 , an irrational number to which no exact, finite closed-form expression is known. On taking into account a suitable search basis, I have found a simple expression for ζ(3) accurate to 21 decimal places, which is triply more accurate than the best previous one. As the short Maple TM code presented here can be easily adapted to study other constants, I decided to supply it to encourage the readers to conduct their own computational experiments, as well as to adopt it in projects of numerical analysis, number theory, or linear algebra. Key words: Integer relation detection, Ap´ery’s constant 2000 MSC: 11Y35, 11Y60, 11M06

“In fact, numerical experimentation is crucial to Number Theory, perhaps more so than to other areas of mathematics. (...) Indeed, as Cassels has said, to a large degree Number Theory is an experimental science.” F. R. Villegas 1. Introduction P∞The1Ap´ery’s constant is defined as the real number to which the infinite series n=1 n3 converges (i.e., 1.2020569 . . .) and it is so designated in honor to R. Ap´ery, who proved in 1978 that this number is irrational [21]. The convergence Email address: [email protected] (F. M. S. Lima)

Preprint submitted to :

Expositiones Mathematicae

August 28, 2012

1 of this series, though slow, by the Cauchy’s integral test, a result P∞ is guaranteed s 1/n for any complex number s with ℜ(s) > 1, that remains valid for n=1 a broader domain in which this series is defined as ζ(s), the Riemann’s zeta function. Ap´ery’s constant can then be identified with ζ(3). For positive integer values of s, s > 1, Euler was the first to derive (in 1734) an exact closed-form expression for ζ(s), namely ζ(2) = π 2 /6, the solution of the Basel problem (see Ref. [2] and references therein). Some years later (the result was found in 1739, but published only in 1750), he succeeded in extending his result to all even values of s [1]:

ζ(2 n) = (−1)n−1

22n−1 B2n 2n π , (2n)!

where n is a positive integer and B2n ∈ Q are Bernoulli numbers. For odd values of s, however, no exact closed-form expression is currently known.2 The increase of interest in ζ(3), which comes from both pure and applied mathematics,3 has stimulated its high-precision numerical computation [18], as well as the search for simple approximate expressions [24]. Let us adopt a reasonable criterium for the adjective “simple,” in the context of finite approximate expressions. Here, it will designate closed-form expressions containing a few terms/factors composed by other known mathematical constants and a few integer numbers with small absolute value. This criterium is, of course, vague due to the forms “a few” and “small”. To make it not-so-imprecise, “a few” will mean less than, say 10, and “small” will mean less than, say 500. A such approximation was presented by Galliani (2002), namely [23]  −3 2 1 1 , (1) + 4 1+2γ − ζ(3) ≈ √ 3 γ π 130 + π 2 where γ is the Euler-Mascheroni constant, which is accurate to 4 decimal places. Another nice simple approximation is r 71 4 γ+ ζ(3) ≈ , (2) 47 due to Hudson (2004), which is accurate to 7 places [23]. Among the many approximations for ζ(3) presented by Hudson, the most accurate is [23] √

ζ(3) ≈ 525587 1/




PN 3 partial sum n=1 1/n , N being any positive integer, of course yields a rational approximation to ζ(3). However, even for N as large as 100 yields only four correct decimal places. 2 In a recent work, it is claimed that ζ(2 n + 1) is not a rational multiple of π 2n+1 , n being a positive integer (see Theorem 22 of Ref. [5]). 3 For instance, given three integers chosen at random, the probability that no common factor will divide them all is 1/ζ(3). Also, if n is a power of 2, then the number #(n) of distinct solutions for n = p + x y with  p prime and x, y positive integers obeys the asymptotic relation #(n)/n ∼ 105 ζ(3)/ 2 π 4 . It also arises in a number of physical problems, including the computation of the electron’s gyromagnetic ratio. 1 The


which yields 12 correct decimal places and, clearly, is not a simple approximate 97525 expression (in our terminology). The same for ζ(3) ≈ 2515594 π 3 , which I have found on searching for a direct integer relation between ζ(3) and π 3 . On trying to reach greater accuracy, however, one soon observes that it is very difficult to avoid the appearance of large integers. In this note, I show how to use PSLQ algorithm for finding the best rational coefficients, within a desired precision, of a null linear combination of a finite number of real constants. As an example, I explore approximate expressions for the Ap´ery’s constant ζ(3), a number to which no closed-form expression in terms of a finite combination of elementary functions of known constants is known. For this, I choose a suitable search basis composed by √ numbers which seem to be closely related to ζ(3), namely π, ln 2 , ln (1 + 2 ), and G (the Catalan’s constant). This yields a simple expression for ζ(3) accurate to 21 decimal places. The short Maple TM code used in this computation is included to stimulate the readers to conduct their own experiments. 2. Searching for integer relations: the PSLQ algorithm An important task in experimental mathematics is to search for integer relations involving a finite set of known numbers. An integer relation algorithm is a computational scheme that, for a given real vector x = (x1 , x2 , . . . , xn ), n > 1, it either finds a nonnull vector of integers a = (a1 , a2 , . . . , an ) such that a1 x1 + a2 x2 + . . . + an xn = 0 or else establishes that there is no such integer vector within a ball of some radius about the origin.4 Presently, the best algorithm for detecting integer relations is the PSLQ algorithm (acronym for Partial Sum of Least sQuares) introduced by Ferguson and Bailey (1992) [13]. A simplified formulation for this algorithm, equivalent to the original one, was subsequently developed by Ferguson and coworkers (1999) [14]. This more efficient version is currently implemented in both Maple TM and Mathematica TM , two of the most popular mathematical softwares. This version of PSLQ, optimized with certain reduction schemes, was named one of the ‘ten algorithms of the century’ in Ref. [3]. In short, PSLQ operates as follows. Given a vector x of n given real numbers, input as a list of floating-point (FP) numbers, the algorithm uses QR decomposition in order to construct a series of matrices Am such that the absolute values of the entries of the vector ym = A−1 m · x decrease monotonically. At any given iteration, the largest and smallest entries of ym usually differ by no more than a few orders of magnitude. When the desired integer relation is detected, the smallest entry of ym abruptly decreases to roughly the computer working precision ǫ and the relation is given by the corresponding column of A−1 m . This numerically stable matrix reduction procedure, together with some techniques that allow machine arithmetic to be used in many intermediate steps, usually

4 The

metric is the Euclidean norm


a21 + a22 + . . . + a2n .


yields a rapid convergence, which makes PSLQ faster than other concurrent algorithms [4]. In fact, if the elements of x are linearly dependent over Q, then PSLQ will find an exact integer relation between them, for a sufficiently precise input. For most applications, high-precision arithmetic is required, which stems from the fact that if one wishes to recover a relation involving n known real numbers, with coefficients accurate to d digits, then the input vector x must be specified to at least n × d digits and one must employ FP arithmetic accurate to at least n × d digits, too. When a relation is detected, the ratio between the smallest and the largest entry of the vector A−1 ·x can be taken as a “confidence level” that the relation is true (i.e., exact) and not an artifact of insufficient numerical precision. Very small ratios at detection certainly indicate the result is probably true.5 3. Closed-form expressions via PSLQ Since an efficient PSLQ routine is available as part of a Maple TM package named IntegerRelations, then simple short codes can be written in this highlevel language for finding approximate expressions for mathematical constants. For illustrating this, let me list the source code I have written for finding an approximate expression for ζ(3). > > >


> >

> > >

Digits := 24: # The number of digits for FP numbers with(IntegerRelations): # Call the package containing PSLQ xSymb:=[Zeta(3),1,Pi^2*ln(2),Pi*ln(2)^2,ln(2)^3,ln(1+sqrt(2))^3,Pi*Catalan]; √ xSymb := [ζ(3), 1, π 2 ln(2), π ln(2)2 , ln(2)3 , ln(1 + 2)3 , π Catalan] n := nops(xSymb); # The number of elements in xSymb n := 7 x:=evalf(xSymb): # Convert to FP numbers a:=PSLQ(x); # Applies PSLQ algorithm to vector x a := [10, 394, −11, 283, −472, −209, −186] soma:=0: for k from 1 to n do soma:=soma+a[k]*xSymb[k]; aprox:=solve(soma=0, Zeta(3));

end do:


√ 11 2 283 236 209 5 93 π Catalan + π ln(2) − π ln(2)2 + ln(2)3 + ln(1 + 2)3 + 197 394 394 197 394 197 evalf( aprox, 30 ); # Our approximation


1.20205690315959428539958993430 evalf( Zeta(3), 30 ); # Exact value (for comparison)

aprox := −

1.20205690315959428539973816151 5 In addition to possessing good numerical stability, PSLQ is guaranteed to find an integer relation in a number of iterations bounded by a polynomial in n.


With this Maple TM routine, I have found the following simple approximate expression for the Ap´ery’s constant: 5 11 2 283 236 3 209 3 93 + π ln 2 − π ln2 2 + ln 2 + α + π G, (4) 197 394 394 197 394 197 √ where α ≡ ln (1 + 2 ), which is accurate to 21 digits, as the reader can check by comparing the last two numerical outputs above (the blue digits are correct). This approximate expression, Eq. (4), is much more accurate than the best previous expressions and this is why it has been included in the popular webpage Wolfram MathWorld [23]. ζ(3) ≈ −

All that rests is to explain the motivation for choosing the vector   √ x = 1, π 2 ln 2, π ln2 2, ln3 2, ln3 (1 + 2 ), π G as the search basis for ζ(3). This comes primarily from a conjecture by Euler (1785) that ζ(3) = a ln3 2 + b π 2 ln 2 , for some a, b ∈ Q [12]. After some experiments with a code similar to the above one, I could not find any such pair of rational coefficients for composing an exact closed-form expression for ζ(3). Not even a simple expression was found with these two terms only.6 I was then inclined to improve the basis by including other weight-3 constants in virtue of their appearance in some exact results involving ζ(3), e.g.:7 (i) Special values of non-elementary functions:   35 1 1 π 2 ln 2+ 16 ln3 2 , ℜ Li3 1+i ln3 2− = 64 ζ(3)+ 48 Li3 21 = 87 ζ(3)− 12 2 5 1 2 1 1 1 2 2 (−4) 2 2 (1) = 2 π ln A+ 12 π ln 2+ 12 π ln π + 8 ζ(3) [22] 192 π ln 2 [7], π ψ (typo corrected); (ii) Infinite series: P∞ (−1)k+1 P∞ ζ(2k) 3 1 1 1 2 1 2 2 k=1 k3 2k (2k) = 4 ζ(3)− 6 ln 2 [15], π k=1 (k+1) 24k = 2 π − 2 π ln 2+ k P∞ (−1)m π2m E2m+1 (1) 35 4 = 31 π 2 ln 2− 23 ζ(3) [10]; m=0 4 ζ(3) − 4π G [8], π (2m+5)!

(iii) Definite integrals: R π/4 R π/4 21 1 x ln (cos x) dx = − 128 ζ(3)+ 81 π G− 32 π 2 ln 2 [20], 0 x2 tan x dx = 0 R π/2 1 π 2 ln 2 [20], 0 x2 / sin x dx = 2 π G − 27 ζ(3) [8], − 21 ζ(3) + 1 π G − 32 R 164(arcsin t)2 4 dt = 14 π 2 ln 2 − 87 ζ(3) (Eq.(6.6.25) in Ref. [6]). t 0 P∞ Here, Li3 (z) := n=1 z n /n3 is the trilogarithm function, ψ (−4) (z) is the polygamma function (extended to negative indexes, according to Ref. [11]), and E2m+1 (x) is the Euler polynomial of degree 2m + 1. 6 I am currently testing a conjecture by Connon (2008) that either a or b could contain a √ factor of 2, or another small surd, or perhaps ln (2π) [9]. 7 The word weight, here, follows the definition introduced by Boros (see Ref. [6], p. 203).


√ With respect to the term with ln3 (1 + 2 ), which is the logarithm of a non-null algebraic number distinct from unit, thus a weight-3 constant, I √ have included it in my search basis because the number arcsinh(1) = ln (1 + 2 ) emerges in the coordinates of the vertices of a cusped hyperbolic cube whose volume is 78 ζ(3), as I have found on exploring certain triple integrals over the unit cube [0, 1]3 , as suggested in the end of Ref. [16].


References [1] T. M. Apostol, Another elementary proof of Eulers formula for ζ(2n), Amer. Math. Monthly 80 (1973) 425–431. [2] R. Ayoub, Euler and the zeta function, Am. Math. Monthly 81 (1974) 1067–1085. [3] D. H. Bailey, Integer relation detection, Computing in Science and Eng. 2 (2000) 24–28. [4] P. Bertok, PSLQ integer relation algorithm. Available at:

\protect\vrule width0pt\protect\href{ [5] G. Bi and Y. Bi, New properties of Fourier series and Riemann zeta function. arXiv:math.AP/1008.5046v4 (2010). [6] G. Boros and V. H. Moll, Irresistible Integrals, Cambridge Univ. Press, New York, 2004. [7] D. J. Broadhurst, Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5). arXiv:math/9803067v1 [math.CA] (1998). [8] Y. J. Cho, M. Jung, J. Choi, and H. M. Srivastava, Closed-form evaluations of definite integrals and associated infinite series involving the Riemann zeta function, Int. J. Comp. Math. 83 (2006) 461–472. [9] D. F. Connon, Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers (Vol. I). arXiv:math.HO/0710.4022v2 (2008). [10] M. J. Dancs and T.-X. He, An Euler-type formula for ζ(2k + 1), J. Number Theory 118 (2006) 192–199. [11] O. Espinosa and V.Moll, A generalized polygamma function, Integral Transforms and Special Functions 15 (2004) 101–115. [12] L. Euler, De relatione inter ternas pluresve quantitates instituenda, Opuscula Analytica 2 (1785) 91–101. [13] H. R. P. Ferguson and D. H. Bailey, A polynomial time, numerically stable integer relation algorithm, Tech. Rep. RNR-91-032, July (1992). [14] H. R. P. Ferguson, D. H. Bailey, and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comput. 68 (1999) 351–369. [15] G. Huvent, Formules d’ordre superieur, Available at:

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[16] F. M. S. Lima, New definite integrals and a two-term dilogarithm identity, Indagat. Math. 23 (2012) 1–9. [17] S. J. Patterson, An introduction to the theory of the Riemann zeta-function (Cambridge, Cambridge Univ. Press, 1988). [18] M. Pr´evost, Series involving zeta and related functions, Appl. Math. Comput. 217 (2011) 5307–5317. ¨ [19] B. Riemann, Uber die Anzahl der Primzahlen unter einer gegebener Gr¨oße, Monatsb. der Berliner Akad. 1858/1860 (1860) 671–680. [20] These expressions were found using Mathematica TM software (release 7). [21] A. van der Poorten, A proof that Euler missed: Ap´ery’s proof of the irrationality of ζ(3), Math. Intelligencer 1 (1979) 196–203. [22] E. W. Weisstein, Ap´ery’s constant. From MathWorld. Available at:

\protect\vrule width0pt\protect\href{}{ [23] E. W. Weisstein, Ap´ery’s constant approximations. From MathWorld. Available at:

\protect\vrule width0pt\protect\href{ [24] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992); Prog. Math., Birkhauser, Basel-Boston, 1994.


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