A013661 Decimal expansion of Pi^2/6 = zeta(2) = Sum_{m>=1} 1/m^2.

1, 6, 4, 4, 9, 3, 4, 0, 6, 6, 8, 4, 8, 2, 2, 6, 4, 3, 6, 4, 7, 2, 4, 1, 5, 1, 6, 6, 6, 4, 6, 0, 2, 5, 1, 8, 9, 2, 1, 8, 9, 4, 9, 9, 0, 1, 2, 0, 6, 7, 9, 8, 4, 3, 7, 7, 3, 5, 5, 5, 8, 2, 2, 9, 3, 7, 0, 0, 0, 7, 4, 7, 0, 4, 0, 3, 2, 0, 0, 8, 7, 3, 8, 3, 3, 6, 2, 8, 9, 0, 0, 6, 1, 9, 7, 5, 8, 7, 0

Offset: 1

Author

N. J. A. Sloane

Keywords

• cons
• nonn
• nice
• changed

Comments

"In 1736 he [Leonard Euler, 1707-1783] discovered the limit to the infinite series, Sum 1/n^2. He did it by doing some rather ingenious mathematics using trigonometric functions that proved the series summed to exactly Pi^2/6. How can this be? ... This demonstrates one of the most startling characteristics of mathematics - the interconnectedness of, seemingly, unrelated ideas." - Clawson [See Hardy and Wright, Theorems 332 and 333. - N. J. A. Sloane, Jan 20 2017]

Also dilogarithm(1). - Rick L. Shepherd, Jul 21 2004

Also Integral_{x>=0} x/(exp(x)-1) dx. [Abramowitz-Stegun, 23.2.7., for s=2, p. 807]

For the partial sums see the fractional sequence A007406/A007407.

Pi^2/6 is also the length of the circumference of a circle whose diameter equals the ratio of volume of an ellipsoid to the circumscribed cuboid. Pi^2/6 is also the length of the circumference of a circle whose diameter equals the ratio of surface area of a sphere to the circumscribed cube. - Omar E. Pol, Oct 07 2011

1 < n^2/(eulerphi(n)*sigma(n)) < zeta(2) for n > 1. - Arkadiusz Wesolowski, Sep 04 2012

Volume of a sphere inscribed in a cube of volume Pi. More generally, Pi^x/6 is the volume of an ellipsoid inscribed in a cuboid of volume Pi^(x-1). - Omar E. Pol, Feb 17 2016

Surface area of a sphere inscribed in a cube of surface area Pi. More generally, Pi^x/6 is the surface area of a sphere inscribed in a cube of surface area Pi^(x-1). - Omar E. Pol, Feb 19 2016

zeta(2)+1 is a weighted average of the integers, n > 2, using zeta(n)-1 as the weights for each n. We have: Sum_{n >= 2} (zeta(n)-1) = 1 and Sum_{n >= 2} n*(zeta(n)-1) = zeta(2)+1. - Richard R. Forberg, Jul 14 2016

zeta(2) is the expected value of sigma(n)/n. - Charlie Neder, Oct 22 2018

Graham shows that a rational number x can be expressed as a finite sum of reciprocals of distinct squares if and only if x is in [0, Pi^2/6-1) U [1, Pi^2/6). See section 4 for other results and Theorem 5 for the underlying principle. - Charles R Greathouse IV, Aug 04 2020

From Peter Bala, Aug 24 2025: (Start)

By definition, zeta(2) = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k^2. The convergence is slow. For example, s(50) = 1.6(25...) is only correct to 1 decimal digit.

Let S(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). It appears that S(n) converges to zeta(2) much more rapidly. For example, S(50) = 1.64493406684822643647241516664602(137...) gives zeta(2) correct to 32 decimal digits. Cf. A073004. (End)

Examples

			1.6449340668482264364724151666460251892189499012067984377355582293700074704032...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.
• F. Aubonnet, D. Guinin and B. Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
• Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 97.
• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
• W. Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington, D.C., 1999, p. xxii.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.4.1 and 5.16, pp. 20, 365.
• Hardy and Wright, 'An Introduction to the Theory of Numbers'. See Theorems 332 and 333.
• A. A. Markoff, Mémoire sur la transformation de séries peu convergentes en séries très convergentes, Mém. de l'Acad. Imp. Sci. de St. Pétersbourg, XXXVII, 1890.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 10, 161-162.
• G. F. Simmons, Calculus Gems, Section B.15, B.24, pp. 270-271, 323-325, McGraw Hill, 1992.
• Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 99, Satz 1.
• A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see p. 261.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• D. H. Bailey, J. M. Borwein, and D. M. Bradley, Experimental determination of Apéry-like identities for zeta(4n+2), arXiv:math/0505270 [math.NT], 2005-2006.
• Peter Bala, New series for old functions.
• Peter Bala, Formulas for A013661.
• David Benko and John Molokach, The Basel Problem as a Rearrangement of Series, The College Mathematics Journal, Vol. 44, No. 3 (May 2013), pp. 171-176.
• R. Calinger, Leonhard Euler: The First St. Petersburg Years (1727-1741), Historia Mathematica, Vol. 23, 1996, pp. 121-166.
• Robin Chapman, Evaluating Zeta(2).
• R. W. Clickery, Probability of two numbers being coprime.
• Alessio Del Vigna, On a solution to the Basel problem based on the fundamental theorem of calculus, arXiv:2104.01710 [math.HO], 2021.
• Leonhard Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
• Leonhard Euler, De summis serierum reciprocarum, E41.
• R. L. Graham, On finite sums of unit fractions, Proceedings of the London Mathematical Society, s3-14 (1964), pp. 193-207. doi:10.1112/plms/s3-14.2.193
• Michael D. Hirschhorn, A simple proof that zeta(2) = Pi^2/6, The Mathematical Intelligencer 33:3 (2011), pp 81-82.
• Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
• Alain Lasjaunias and Jean-Paul Tran, A note on the equality Pi^2/6 = Sum_{n>=1} 1/n^2, arXiv:2312.02245 [math.HO], 2023.
• Math. Reference Project, The Zeta Function, Zeta(2).
• Math. Reference Project, The Zeta Function, Odds That Two Numbers Are Coprime".
• Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
• Jon Perry, Prime Product Paradox
• Simon Plouffe, Plouffe's Inverter, Zeta(2) or Pi**2/6 to 100000 digits.
• Simon Plouffe, Zeta(2) or Pi**2/6 to 10000 places.
• Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each (gzipped file)
• A. L. Robledo, value of the Riemann zeta function at s=2, PlanetMath.org.
• E. Sandifer, How Euler Did It, Estimating the Basel Problem.
• E. Sandifer, How Euler Did It, Basel Problem with Integrals.
• C. Tooth, Pi squared over six.
• Eric Weisstein's World of Mathematics, Dilogarithm.
• Eric Weisstein's World of Mathematics, Riemann Zeta Function zeta(2).
• Wikipedia, Basel Problem.
• Wikipedia, Bailey-Borwein-Plouffe formula.
• Herbert S. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics & Theoretical Computer Science, Vol 3, No 4 (1999).
• Index entries for transcendental numbers.
• Index entries for zeta function.

Crossrefs

Cf. A001008 (H(n): numerators), A002805 (denominators), A013679 (continued fraction), A002117 (zeta(3)), A013631 (cont.frac. for zeta(3)), A013680 (cont.frac. for zeta(4)), 1/A059956, A108625, A142995, A142999.

Cf. A000290.

Programs

Magma

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi^2/6))); // Vincenzo Librandi, Oct 13 2015

Maple

evalf(Pi^2/6,120); # Muniru A Asiru, Oct 25 2018
# Calculates an approximation with n exact decimal places (small deviation
# in the last digits are possible). Goes back to ideas of A. A. Markoff 1890.
zeta2 := proc(n) local q, s, w, v, k; q := 0; s := 0; w := 1; v := 4;
for k from 2 by 2 to 7*n/2 do
    w := w*v/k;
    q := q + v;
    v := v + 8;
    s := s + 1/(w*q);
od; 12*s; evalf[n](%) end:
zeta2(1000); # Peter Luschny, Jun 10 2020

Mathematica

RealDigits[N[Pi^2/6, 100]][[1]]
RealDigits[Zeta[2],10,120][[1]] (* Harvey P. Dale, Jan 08 2021 *)

Maxima

fpprec : 100$ ev(bfloat(zeta(2)))$ bfloat(%); /* Martin Ettl, Oct 21 2012 */

PARI

default(realprecision, 200); Pi^2/6

PARI

default(realprecision, 200); dilog(1)

PARI

default(realprecision, 200); zeta(2)

PARI

A013661(n)={localprec(n+2); Pi^2/.6\10^n%10} \\ Corrected and improved by M. F. Hasler, Apr 20 2021

PARI

default(realprecision, 20080); x=Pi^2/6; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b013661.txt", n, " ", d)); \\ Harry J. Smith, Apr 29 2009

PARI

sumnumrat(1/x^2, 1) \\ Charles R Greathouse IV, Jan 20 2022

Python

# Use some guard digits when computing.
# BBP formula (3 / 16) P(2, 64, 6, (16, -24, -8, -6,  1, 0)).
from decimal import Decimal as dec, getcontext
def BBPzeta2(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(64)
    for k in range(int(n * 0.5536546824812272) + 1):
        sixk = dec(6 * k)
        s += f * ( dec(16) / (sixk + 1) ** 2 - dec(24) / (sixk + 2) ** 2
                 - dec(8)  / (sixk + 3) ** 2 - dec(6)  / (sixk + 4) ** 2
                 + dec(1)  / (sixk + 5) ** 2 )
        f /= g
    return (s * dec(3)) / dec(16)
print(BBPzeta2(2000))  # Peter Luschny, Nov 01 2023

Formula

Limit_{n->oo} (1/n)*(Sum_{k=1..n} frac((n/k)^(1/2))) = zeta(2) and in general we have lim_{n->oo} (1/n)*(Sum_{k=1..n} frac((n/k)^(1/m))) = zeta(m), m >= 2. - Yalcin Aktar, Jul 14 2005

Equals Integral_{x=0..1} (log(x)/(x-1)) dx or Integral_{x>=1} (log(x/(x-1))/x) dx. - Jean-François Alcover, May 30 2013

For s >= 2 (including Complex), zeta(s) = Product_{n >= 1} prime(n)^s/(prime(n)^s - 1). - Fred Daniel Kline, Apr 10 2014

Also equals 1 + Sum_{n>=0} (-1)^n*StieltjesGamma(n)/n!. - Jean-François Alcover, May 07 2014

zeta(2) = Sum_{n>=1} ((floor(sqrt(n)) - floor(sqrt(n-1)))/n). - Mikael Aaltonen, Jan 10 2015

zeta(2) = Sum_{n>=1} (((sqrt(5)-1)/2/sqrt(5))^n/n^2) + Sum_{n>=1} (((sqrt(5)+1)/2/sqrt(5))^n/ n^2) + log((sqrt(5)-1)/2/sqrt(5))log((sqrt(5)+1)/2/sqrt(5)). - Seiichi Kirikami, Oct 14 2015

The above formula can also be written zeta(2) = dilog(x) + dilog(y) + log(x)*log(y) where x = (1-1/sqrt(5))/2 and y=(1+1/sqrt(5))/2. - Peter Luschny, Oct 16 2015

zeta(2) = Integral_{x>=0} 1/(1 + e^x^(1/2)) dx, because (1 - 1/2^(s-1))*Gamma[1 + s]*Zeta[s] = Integral_{x>=0} 1/(1 + e^x^(1/s)) dx. After Jean-François Alcover in A002162. - Mats Granvik, Sep 12 2016

zeta(2) = Product_{n >= 1} (144*n^4)/(144*n^4 - 40*n^2 + 1). - Fred Daniel Kline, Oct 29 2016

zeta(2) = lim_{n->oo} (1/n) * Sum_{k=1..n} A017665(k)/A017666(k). - Dimitri Papadopoulos, May 10 2019 [See the Walfisz reference, and a comment in A284648, citing also the Sándor et al. Handbook. - Wolfdieter Lang, Aug 22 2019]

Equals Sum_{k>=1} H(k)/(k*(k+1)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 16 2020

Equals (8/3)*(1/2)!^4 = (8/3)*Gamma(3/2)^4. - Gary W. Adamson, Aug 17 2021

Equals ((m+1)/m) * Integral_{x=0..1} log(Sum {k=0..m} x^k )/x dx, m > 0 (Aubonnet reference). - _Bernard Schott, Feb 11 2022

Equals 1 + Sum_{n>=1} n*(zeta(n+2)-1). - Richard R. Forberg, Jun 04 2023; improved by Natalia L. Skirrow, Jul 25 2025

Equals Psi'(1) where Psi'(x) is the trigamma function (by Abramowitz Stegun 6.4.2). - Andrea Pinos, Oct 22 2024

Equals Integral_{x=0..1} Integral_{y=0..1} 1/(1 - x*y) dy dx. - Kritsada Moomuang, May 22 2025

Equals 1 + Sum_{n>=1} 1/(n^2*(n+1)). - Natalia L. Skirrow, Jul 25 2025

Extensions

Edited by N. J. A. Sloane, Nov 22 2023