1,2

"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)

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.

Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See pp. 237, 266-267.

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.

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.

Ryan Goulden, Closed-Form Evaluation of Arctanh Power Sums via Infinite Products, arXiv:2602.06244 [math.GM], 2026. See p. 17.

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.

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

Equals 3*Sum_{k>0} (k - 1)!^2/(2*k)! [Knopp]. - Stefano Spezia, Jan 31 2026

Equals lim_{m->oo} (1/m^2) * Sum_{i,j=1..m} A000005(gcd(i, j)). - Amiram Eldar, Mar 14 2026

1.6449340668482264364724151666460251892189499012067984377355582293700074704032...

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

RealDigits[N[Pi^2/6, 100]][[1]]

RealDigits[Zeta[2], 10, 120][[1]] (* Harvey P. Dale, Jan 08 2021 *)

(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

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

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

(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

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. A000005, A000290.

Sequence in context: A201587 A110756 A200698 * A209273 A330934 A019174

Adjacent sequences: A013658 A013659 A013660 * A013662 A013663 A013664

cons,nonn,nice

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

approved