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 A013661 Decimal expansion of zeta(2) = Pi^2/6. 103
 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} 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 [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. 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 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. Hardy and Wright, 'An Introduction to the Theory of Numbers'. See Theorems 332 and 333. 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, 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 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. 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 [math.HO], 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 Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each (gzipped file) 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 Wikipedia, Basel Problem H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics & Theoretical Computer Science, Vol 3, No 4 (1999). FORMULA 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 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 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) = Integral_{x = -inf..inf} x^2*sech^2(x) dx. _ Peter Bala, Sep 21 2016 zeta(2) = Product_{n >=1} (144*n^4)/(144*n^4 - 40*n^2 + 1). - Fred Daniel Kline, Oct 29 2016 EXAMPLE 1.6449340668482264364724151666460251892189499012067984377355582293700074704032... MAPLE evalf(Pi^2/6, 120); # Muniru A Asiru, Oct 25 2018 MATHEMATICA RealDigits[N[Pi^2/6, 100]][[1]] PROG (PARI) default(realprecision, 200); Pi^2/6 (PARI) default(realprecision, 200); dilog(1) (PARI) default(realprecision, 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, A002117, 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 STATUS approved

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Last modified November 19 17:03 EST 2018. Contains 317354 sequences. (Running on oeis4.)