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A013661 Decimal expansion of zeta(2) = Pi^2/6. 89
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sum_{m = 1..inf } 1/m^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

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

Also Integral_{x=0..inf} x/(exp(x)-1).

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

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.

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 97.

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington, D.C., 1999, p. xxii.

G. F. Simmons, Calculus Gems, Section B.15,B.24 pp. 270-1,323-5 McGraw Hill 1992.

A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, 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 Ap'ery-like identities for zeta(4n+2), arXiv:math/0505270 [math.NT], 2005-2006.

P. Bala, New series for old functions

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, Leonard Euler: The First St. Petersburg Years (1727-1741), Historia Mathematica, Vol. 23, 1996, pp. 121-166.

R. Chapman, Evaluating Zeta(2):14 Proofs to Zeta(2)= (pi)^2/6

R. W. Clickery, Probability of two numbers being coprime [Broken link]

L. Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.

L. Euler, De summis serierum reciprocarum, E41.

Michael D. Hirschhorn, A simple proof that zeta(2) = Pi^2/6, The Mathematical Intelligencer 33:3 (2011), pp 81-82.

Math. Reference Project, The Zeta Function, Zeta(2)

Math. Reference Project, The Zeta Function, Odds That Two Numbers Are Coprime"

R. 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, 2012.

J. Perry, Prime Product Paradox [White page?]

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

A. L. Robledo, PlanetMath.org, value of the Riemann zeta function at s=2

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, Riemann Zeta Function 2

Eric Weisstein's World of Mathematics, Dilogarithm MathWorld page

H. Wilf, Accelerated series for universal constants, by the WZ method

Index entries for zeta function.

FORMULA

Lim_(n->inf) of (1/n)*(Sum_{k=1..n} frac((n/k)^(1/2))) = zeta(2) and in general we have Lim_{n->infinity} (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

From Peter Bala, Dec 01 2013: (Start)

Lim_{n->inf} Sum_{k=1..n-1} (log(n) - log(k))/(n - k).

Also integral_{x=0..1} z^(z^(z^(...))) dx, where z = x^(-x). (End)

From Peter Bala, Dec 10 2013: (Start)

zeta(2) = (16/9)*Sum_{n even} n^2*(n^2 + 1)/(n^2 - 1)^3.

zeta(2) = 3*Sum_{n >= 1} (20*n^2 - 8*n + 1)/( ((2*n)*(2*n - 1))^2*C(4*n,2*n) ).

zeta(2) = 3*Sum_{n >= 1} (1701*n^4 - 1944*n^3 + 729*n^2 - 96*n + 4)/( ((3*n)*(3*n - 1)*(3*n - 2))^2*C(6*n,3*n) ) (Bala, Section 6).

See A108625 for series and continued fraction expansions for zeta(2) associated with the crystal ball sequences for the A_n lattice. See also A142995 and A142999. (End)

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

EXAMPLE

1.6449340668482264364724151666460251892189499012067984377355582293700074704032...

MATHEMATICA

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

PROG

(PARI) \p 200; Pi^2/6

(PARI) \p 200 dilog(1) \p 200 zeta(2)

(PARI) a(n)=if(n<1, 0, default(realprecision, n+2); floor(Pi^2/6*10^(n-1))%10)

(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

(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

CROSSREFS

Cf. A013679, A013631, A013680, 1/A059956, A108625, A142995, A142999.

Sequence in context: A201587 A110756 A200698 * A209273 A019174 A019166

Adjacent sequences:  A013658 A013659 A013660 * A013662 A013663 A013664

KEYWORD

cons,nonn,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 1 14:41 EDT 2016. Contains 274321 sequences.