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 A013661 Decimal expansion of zeta(2) = Pi^2/6. 103

%I

%S 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,

%T 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,

%U 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

%N Decimal expansion of zeta(2) = Pi^2/6.

%C Sum_{m>=1} 1/m^2.

%C "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]

%C Also dilogarithm(1). - _Rick L. Shepherd_, Jul 21 2004

%C Also Integral_{x>=0} x/(exp(x)-1) dx.

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

%C 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

%C 1 < n^2/(eulerphi(n)*sigma(n)) < zeta(2) for n > 1. - _Arkadiusz Wesolowski_, Sep 04 2012

%C 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

%C 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

%C 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

%C zeta(2) is the expected value of sigma(n)/n. - _Charlie Neder_, Oct 22 2018

%D 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.

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

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

%D Hardy and Wright, 'An Introduction to the Theory of Numbers'. See Theorems 332 and 333.

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

%D A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see p. 261.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.

%H Harry J. Smith, <a href="/A013661/b013661.txt">Table of n, a(n) for n = 1..20000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H D. H. Bailey, J. M. Borwein and D. M. Bradley, <a href="https://arxiv.org/abs/math/0505270">Experimental determination of Ap'ery-like identities for zeta(4n+2)</a>, arXiv:math/0505270 [math.NT], 2005-2006.

%H P. Bala, <a href="/A002117/a002117.pdf">New series for old functions</a>

%H David Benko and John Molokach, <a href="http://www.jstor.org/stable/10.4169/college.math.j.44.3.171">The Basel Problem as a Rearrangement of Series</a>, The College Mathematics Journal, Vol. 44, No. 3 (May 2013), pp. 171-176.

%H R. Calinger, <a href="http://dx.doi.org/10.1006/hmat.1996.0015">Leonard Euler: The First St. Petersburg Years (1727-1741)</a>, Historia Mathematica, Vol. 23, 1996, pp. 121-166.

%H R. Chapman, <a href="http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf">Evaluating Zeta(2):14 Proofs to Zeta(2)= (pi)^2/6</a>

%H R. W. Clickery, <a href="http://www.cacr.caltech.edu/~roy/upi/coprime.html">Probability of two numbers being coprime</a> [Broken link]

%H L. Euler, <a href="http://arXiv.org/abs/math.HO/0506415">On the sums of series of reciprocals</a>, arXiv:math/0506415 [math.HO], 2005-2008.

%H L. Euler, <a href="http://eulerarchive.maa.org/pages/E041.html">De summis serierum reciprocarum</a>, E41.

%H Michael D. Hirschhorn, <a href="http://dx.doi.org/10.1007/s00283-011-9217-4">A simple proof that zeta(2) = Pi^2/6</a>, The Mathematical Intelligencer 33:3 (2011), pp 81-82.

%H Math. Reference Project, <a href="http://www.mathreference.com/lc-z,zeta2.html">The Zeta Function, Zeta(2)</a>

%H Math. Reference Project, <a href="http://www.mathreference.com/lc-z,cop.html">The Zeta Function, Odds That Two Numbers Are Coprime"</a>

%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012.

%H Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/zeta2.txt">Zeta(2) or Pi**2/6 to 100000 digits</a>

%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap96.html">Zeta(2) or Pi**2/6 to 10000 places</a>

%H Simon Plouffe, <a href="/A293904/a293904_4096.gz">Zeta(2) to Zeta(4096) to 2048 digits each</a> (gzipped file)

%H A. L. Robledo, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/ValueOfTheRiemannZetaFunctionAtS2.html">value of the Riemann zeta function at s=2</a>

%H E. Sandifer, How Euler Did It, <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2002%20Estimating%20the%20Basel%20Problem.pdf">Estimating the Basel Problem</a>

%H E. Sandifer, How Euler Did It, <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2005%20Basel%20with%20integrals.pdf">Basel Problem with Integrals</a>

%H C. Tooth, <a href="http://www.pisquaredoversix.force9.co.uk">Pi squared over six</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html">Riemann Zeta Function 2</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm MathWorld page</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Basel_problem">Basel Problem</a>

%H H. Wilf, <a href="https://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/108.1.html">Accelerated series for universal constants, by the WZ method</a>, Discrete Mathematics & Theoretical Computer Science, Vol 3, No 4 (1999).

%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>.

%F Lim_{n->infinity} (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

%F 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

%F From _Peter Bala_, Dec 01 2013: (Start)

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

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

%F From _Peter Bala_, Dec 10 2013: (Start)

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

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

%F 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).

%F 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)

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

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

%F zeta(2) = Sum_{n>=1} ((floor(sqrt(n)) - floor(sqrt(n-1)))/n). - _Mikael Aaltonen_, Jan 10 2015

%F 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

%F 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

%F 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

%F zeta(2) = Integral_{x = -inf..inf} x^2*sech^2(x) dx. _ _Peter Bala_, Sep 21 2016

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

%e 1.6449340668482264364724151666460251892189499012067984377355582293700074704032...

%p evalf(Pi^2/6,120); # _Muniru A Asiru_, Oct 25 2018

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

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

%o (PARI) default(realprecision, 200); dilog(1)

%o (PARI) default(realprecision, 200); zeta(2)

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

%o (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

%o (Maxima) fpprec : 100\$ ev(bfloat(zeta(2)))\$ bfloat(%); /* _Martin Ettl_, Oct 21 2012 */

%o (MAGMA) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi^2/6))); // _Vincenzo Librandi_, Oct 13 2015

%Y Cf. A013679, A002117, A013631, A013680, 1/A059956, A108625, A142995, A142999.

%K cons,nonn,nice,changed

%O 1,2

%A _N. J. A. Sloane_

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Last modified November 14 01:55 EST 2018. Contains 317159 sequences. (Running on oeis4.)