--------------------------------------- Explicit Expressions for the Expectation, Variance and the Moments about the\ mean up to the, 8, -th As well as their asymptotic to order, 2 By Shalosh B. Ekhad Let n - 1 ----- \ 1 Hn[r](n) = ) ---- / r ----- i i = 1 In particular, Hn[1](n) is the (n-1)-th Harmonic number. Note that Hn[r](n) is the partial sum, up to n-1, of Zeta[r] Consider the absent-minded passenger problem, where there are n passengers, \ the FIRST of whom is absent-minded Let X be the random variable: Number of Passengers sitting in the wrong seat We have the following, 8, theorems each of them with its own corollary Theorem 1: The Expectation of X is Hn[1] and in Maple notation Hn[1] Corollary 1: The asymptotics of the expectation to order, 2, is 1 1 ln(n) + gamma - --- + O(----) 2 n 2 n Theorem 2: The Variance of X is n Hn[1] - n Hn[2] + 2 Hn[1] --------------------------- n and in Maple notation (n*Hn[1]-n*Hn[2]+2*Hn[1])/n Corollary 2: The asymptotics of the variance to order, 2, is 2 Pi 1/2 + 2 ln(n) + 2 gamma 1 ln(n) + gamma - --- + ----------------------- + O(----) 6 n 2 n and in Maple notation ln(n)+gamma-1/6*Pi^2+(1/2+2*ln(n)+2*gamma)/n+O(1/n^2) Theorem Number, 3, : The , 3, -th moment about the mean is 2 n Hn[1] - 3 n Hn[2] + 2 n Hn[3] - 3 Hn[1] + 6 Hn[1] - 3 Hn[2] -------------------------------------------------------------- n and in Maple notation (n*Hn[1]-3*n*Hn[2]+2*n*Hn[3]-3*Hn[1]^2+6*Hn[1]-3*Hn[2])/n Corollary Number, 3, : The asymptotics of the variance to order, 2, is 2 Pi ln(n) + gamma - --- + 2 Zeta(3) 2 2 2 Pi 5/2 - 3 (ln(n) + gamma) + 6 ln(n) + 6 gamma - --- 2 1 + -------------------------------------------------- + O(----) n 2 n and in Maple notation ln(n)+gamma-1/2*Pi^2+2*Zeta(3)+(5/2-3*(ln(n)+gamma)^2+6*ln(n)+6*gamma-1/2*Pi^2) /n+O(1/n^2) and in floating-point ln(n)-1.953472731+(1.028491787-3.*(ln(n)+.5772156649)^2+6.*ln(n))/n+O(1/n^2) Theorem Number, 4, : The , 4, -th moment about the mean is 2 2 3 (3 n Hn[1] - 6 n Hn[1] Hn[2] + 3 n Hn[2] + 4 Hn[1] + n Hn[1] - 7 n Hn[2] 2 + 12 n Hn[3] - 6 n Hn[4] - 6 Hn[1] + 14 Hn[1] - 18 Hn[2] + 8 Hn[3])/n and in Maple notation (3*n*Hn[1]^2-6*n*Hn[1]*Hn[2]+3*n*Hn[2]^2+4*Hn[1]^3+n*Hn[1]-7*n*Hn[2]+12*n*Hn[3] -6*n*Hn[4]-6*Hn[1]^2+14*Hn[1]-18*Hn[2]+8*Hn[3])/n Corollary Number, 4, : The asymptotics of the variance to order, 2, is 4 2 2 2 Pi 7 Pi 3 (ln(n) + gamma) - (ln(n) + gamma) Pi + --- + ln(n) + gamma - ----- 60 6 / 2 | 7 Pi 3 + 12 Zeta(3) + |17 ln(n) + 17 gamma - ----- + 4 (ln(n) + gamma) + 13/2 \ 2 \ 2 | 1 - 6 (ln(n) + gamma) + 8 Zeta(3)|/n + O(----) / 2 n and in Maple notation 3*(ln(n)+gamma)^2-(ln(n)+gamma)*Pi^2+1/60*Pi^4+ln(n)+gamma-7/6*Pi^2+12*Zeta(3)+ (17*ln(n)+17*gamma-7/2*Pi^2+4*(ln(n)+gamma)^3+13/2-6*(ln(n)+gamma)^2+8*Zeta(3)) /n+O(1/n^2) and in floating-point 3.*(ln(n)+.5772156649)^2-8.869604404*ln(n)-.58604538+(17.*ln(n)-8.614493886+4.* (ln(n)+.5772156649)^3-6.*(ln(n)+.5772156649)^2)/n+O(1/n^2) Theorem Number, 5, : The , 5, -th moment about the mean is 4 2 2 (-5 Hn[1] + 10 n Hn[1] - 40 n Hn[1] Hn[2] + 20 n Hn[1] Hn[3] + 30 n Hn[2] 3 - 20 n Hn[2] Hn[3] + 10 Hn[1] + n Hn[1] - 15 n Hn[2] + 50 n Hn[3] 2 2 - 60 n Hn[4] + 24 n Hn[5] + 5 Hn[1] - 30 Hn[1] Hn[2] + 15 Hn[2] + 30 Hn[1] - 75 Hn[2] + 80 Hn[3] - 30 Hn[4])/n and in Maple notation (-5*Hn[1]^4+10*n*Hn[1]^2-40*n*Hn[1]*Hn[2]+20*n*Hn[1]*Hn[3]+30*n*Hn[2]^2-20*n*Hn [2]*Hn[3]+10*Hn[1]^3+n*Hn[1]-15*n*Hn[2]+50*n*Hn[3]-60*n*Hn[4]+24*n*Hn[5]+5*Hn[1 ]^2-30*Hn[1]*Hn[2]+15*Hn[2]^2+30*Hn[1]-75*Hn[2]+80*Hn[3]-30*Hn[4])/n Corollary Number, 5, : The asymptotics of the variance to order, 2, is 2 2 10 (ln(n) + gamma) - 20/3 (ln(n) + gamma) Pi + 20 (ln(n) + gamma) Zeta(3) 4 2 Pi 2 5 Pi + --- - 10/3 Pi Zeta(3) + ln(n) + gamma - ----- + 50 Zeta(3) + 24 Zeta(5) 6 2 / 2 | 4 115 Pi + |-5 (ln(n) + gamma) + 60 ln(n) + 60 gamma - ------- + 90 Zeta(3) \ 6 3 2 2 + 10 (ln(n) + gamma) + 29/2 + 5 (ln(n) + gamma) - 5 (ln(n) + gamma) Pi 4\ Pi | 1 + ---|/n + O(----) 12 / 2 n and in Maple notation 10*(ln(n)+gamma)^2-20/3*(ln(n)+gamma)*Pi^2+20*(ln(n)+gamma)*Zeta(3)+1/6*Pi^4-10 /3*Pi^2*Zeta(3)+ln(n)+gamma-5/2*Pi^2+50*Zeta(3)+24*Zeta(5)+(-5*(ln(n)+gamma)^4+ 60*ln(n)+60*gamma-115/6*Pi^2+90*Zeta(3)+10*(ln(n)+gamma)^3+29/2+5*(ln(n)+gamma) ^2-5*(ln(n)+gamma)*Pi^2+1/12*Pi^4)/n+O(1/n^2) and in floating-point 10.*(ln(n)+.5772156649)^2-40.75622464*ln(n)+13.47873047+(-5.*(ln(n)+.5772156649 )^4+10.65197798*ln(n)-52.21638368+10.*(ln(n)+.5772156649)^3+5.*(ln(n)+.57721566\ 49)^2)/n+O(1/n^2) Theorem Number, 6, : The , 6, -th moment about the mean is 5 3 2 2 3 (6 Hn[1] + 15 n Hn[1] - 45 n Hn[1] Hn[2] + 45 n Hn[1] Hn[2] - 15 n Hn[2] 4 2 - 15 Hn[1] + 25 n Hn[1] - 180 n Hn[1] Hn[2] + 220 n Hn[1] Hn[3] 2 - 90 n Hn[1] Hn[4] + 195 n Hn[2] - 300 n Hn[2] Hn[3] + 90 n Hn[2] Hn[4] 2 3 + 40 n Hn[3] + 20 Hn[1] + n Hn[1] - 31 n Hn[2] + 180 n Hn[3] 2 - 390 n Hn[4] + 360 n Hn[5] - 120 n Hn[6] + 90 Hn[1] - 330 Hn[1] Hn[2] 2 + 120 Hn[1] Hn[3] + 225 Hn[2] - 120 Hn[2] Hn[3] + 62 Hn[1] - 270 Hn[2] + 520 Hn[3] - 450 Hn[4] + 144 Hn[5])/n and in Maple notation (6*Hn[1]^5+15*n*Hn[1]^3-45*n*Hn[1]^2*Hn[2]+45*n*Hn[1]*Hn[2]^2-15*n*Hn[2]^3-15* Hn[1]^4+25*n*Hn[1]^2-180*n*Hn[1]*Hn[2]+220*n*Hn[1]*Hn[3]-90*n*Hn[1]*Hn[4]+195*n *Hn[2]^2-300*n*Hn[2]*Hn[3]+90*n*Hn[2]*Hn[4]+40*n*Hn[3]^2+20*Hn[1]^3+n*Hn[1]-31* n*Hn[2]+180*n*Hn[3]-390*n*Hn[4]+360*n*Hn[5]-120*n*Hn[6]+90*Hn[1]^2-330*Hn[1]*Hn [2]+120*Hn[1]*Hn[3]+225*Hn[2]^2-120*Hn[2]*Hn[3]+62*Hn[1]-270*Hn[2]+520*Hn[3]-\ 450*Hn[4]+144*Hn[5])/n Corollary Number, 6, : The asymptotics of the variance to order, 2, is 2 2 4 2 -15/2 (ln(n) + gamma) Pi + 1/4 (ln(n) + gamma) Pi - 30 (ln(n) + gamma) Pi 6 2 5 Pi + 220 (ln(n) + gamma) Zeta(3) - 50 Pi Zeta(3) - ----- 168 4 3 2 13 Pi 2 + 15 (ln(n) + gamma) + 25 (ln(n) + gamma) + ------ + 40 Zeta(3) 12 2 / 31 Pi | + 180 Zeta(3) + 360 Zeta(5) + ln(n) + gamma - ------ + | 6 \ 4 5 2 -15 (ln(n) + gamma) + 6 (ln(n) + gamma) + 61/2 - 20 Pi Zeta(3) + 144 Zeta(5) + 710 Zeta(3) + 120 (ln(n) + gamma) Zeta(3) 2 + 15 (ln(n) + gamma) (-ln(n) - gamma) + 255/2 (ln(n) + gamma) 2 2 3 - 15/2 (-ln(n) - gamma) Pi - 70 (ln(n) + gamma) Pi + 20 (ln(n) + gamma) 4 \ 2 11 Pi | 1 - 95 Pi + ------ + 217 ln(n) + 217 gamma|/n + O(----) 8 / 2 n and in Maple notation -15/2*(ln(n)+gamma)^2*Pi^2+1/4*(ln(n)+gamma)*Pi^4-30*(ln(n)+gamma)*Pi^2+220*(ln (n)+gamma)*Zeta(3)-50*Pi^2*Zeta(3)-5/168*Pi^6+15*(ln(n)+gamma)^3+25*(ln(n)+ gamma)^2+13/12*Pi^4+40*Zeta(3)^2+180*Zeta(3)+360*Zeta(5)+ln(n)+gamma-31/6*Pi^2+ (-15*(ln(n)+gamma)^4+6*(ln(n)+gamma)^5+61/2-20*Pi^2*Zeta(3)+144*Zeta(5)+710* Zeta(3)+120*(ln(n)+gamma)*Zeta(3)+15*(ln(n)+gamma)*(-ln(n)-gamma)+255/2*(ln(n)+ gamma)^2-15/2*(-ln(n)-gamma)*Pi^2-70*(ln(n)+gamma)*Pi^2+20*(ln(n)+gamma)^3-95* Pi^2+11/8*Pi^4+217*ln(n)+217*gamma)/n+O(1/n^2) and in floating-point -49.02203303*(ln(n)+.5772156649)^2-6.2833406*ln(n)+76.56450421+15.*(ln(n)+.5772\ 156649)^3+(-15.*(ln(n)+.5772156649)^4+6.*(ln(n)+.5772156649)^5-155.2117560-255.\ 6034469*ln(n)+15.*(ln(n)+.5772156649)*(-1.*ln(n)-.5772156649)+127.5000000*(ln(n )+.5772156649)^2+20.*(ln(n)+.5772156649)^3)/n+O(1/n^2) Theorem Number, 7, : The , 7, -th moment about the mean is 6 5 3 2 2 (-7 Hn[1] + 21 Hn[1] + 105 n Hn[1] - 525 n Hn[1] Hn[2] + 210 n Hn[1] Hn[3] 2 3 + 735 n Hn[1] Hn[2] - 420 n Hn[1] Hn[2] Hn[3] - 315 n Hn[2] 2 4 2 + 210 n Hn[2] Hn[3] - 35 Hn[1] + 56 n Hn[1] - 686 n Hn[1] Hn[2] + 1540 n Hn[1] Hn[3] - 1470 n Hn[1] Hn[4] + 504 n Hn[1] Hn[5] 2 + 1050 n Hn[2] - 2800 n Hn[2] Hn[3] + 1890 n Hn[2] Hn[4] 2 3 - 504 n Hn[2] Hn[5] + 840 n Hn[3] - 420 n Hn[3] Hn[4] + 140 Hn[1] 2 2 3 - 315 Hn[1] Hn[2] + 315 Hn[1] Hn[2] - 105 Hn[2] + n Hn[1] - 63 n Hn[2] + 602 n Hn[3] - 2100 n Hn[4] + 3360 n Hn[5] - 2520 n Hn[6] + 720 n Hn[7] 2 + 469 Hn[1] - 2310 Hn[1] Hn[2] + 1960 Hn[1] Hn[3] - 630 Hn[1] Hn[4] 2 2 + 2100 Hn[2] - 2520 Hn[2] Hn[3] + 630 Hn[2] Hn[4] + 280 Hn[3] + 126 Hn[1] - 903 Hn[2] + 2800 Hn[3] - 4200 Hn[4] + 3024 Hn[5] - 840 Hn[6] )/n and in Maple notation (-7*Hn[1]^6+21*Hn[1]^5+105*n*Hn[1]^3-525*n*Hn[1]^2*Hn[2]+210*n*Hn[1]^2*Hn[3]+ 735*n*Hn[1]*Hn[2]^2-420*n*Hn[1]*Hn[2]*Hn[3]-315*n*Hn[2]^3+210*n*Hn[2]^2*Hn[3]-\ 35*Hn[1]^4+56*n*Hn[1]^2-686*n*Hn[1]*Hn[2]+1540*n*Hn[1]*Hn[3]-1470*n*Hn[1]*Hn[4] +504*n*Hn[1]*Hn[5]+1050*n*Hn[2]^2-2800*n*Hn[2]*Hn[3]+1890*n*Hn[2]*Hn[4]-504*n* Hn[2]*Hn[5]+840*n*Hn[3]^2-420*n*Hn[3]*Hn[4]+140*Hn[1]^3-315*Hn[1]^2*Hn[2]+315* Hn[1]*Hn[2]^2-105*Hn[2]^3+n*Hn[1]-63*n*Hn[2]+602*n*Hn[3]-2100*n*Hn[4]+3360*n*Hn [5]-2520*n*Hn[6]+720*n*Hn[7]+469*Hn[1]^2-2310*Hn[1]*Hn[2]+1960*Hn[1]*Hn[3]-630* Hn[1]*Hn[4]+2100*Hn[2]^2-2520*Hn[2]*Hn[3]+630*Hn[2]*Hn[4]+280*Hn[3]^2+126*Hn[1] -903*Hn[2]+2800*Hn[3]-4200*Hn[4]+3024*Hn[5]-840*Hn[6])/n Corollary Number, 7, : The asymptotics of the variance to order, 2, is 4 2 7/6 Pi Zeta(3) + 504 (ln(n) + gamma) Zeta(5) - 84 Pi Zeta(5) 2 + 210 (ln(n) + gamma) Zeta(3) + 1540 (ln(n) + gamma) Zeta(3) 2 2 49 4 - 175/2 (ln(n) + gamma) Pi + -- (ln(n) + gamma) Pi 12 2 2 - 343/3 (ln(n) + gamma) Pi + 56 (ln(n) + gamma) + 720 Zeta(7) 6 2 3 5 Pi - 1400/3 Pi Zeta(3) + 3360 Zeta(5) + 105 (ln(n) + gamma) - ----- 8 2 4 2 21 Pi 35 Pi + 602 Zeta(3) + 840 Zeta(3) - ------ + ------ 2 6 / 2 | - 70 (ln(n) + gamma) Pi Zeta(3) + ln(n) + gamma + | \ 210 (-ln(n) - gamma) Zeta(3) + 1960 (ln(n) + gamma) Zeta(3) / 2\ | Pi | 2 2 - 420 |-ln(n) - gamma - ---| Zeta(3) - 105/2 (ln(n) + gamma) Pi \ 12 / 2 4 - 175/2 (-ln(n) - gamma) Pi + 7/4 (ln(n) + gamma) Pi 2 2 - 630 (ln(n) + gamma) Pi + 1883/2 (ln(n) + gamma) 4 + 105 (ln(n) + gamma) (-ln(n) - gamma) + 125/2 - 35 (ln(n) + gamma) 5 2 + 21 (ln(n) + gamma) - 490 Pi Zeta(3) + 3276 Zeta(5) 6 3 5 Pi 6 + 140 (ln(n) + gamma) - ----- - 7 (ln(n) + gamma) + 4830 Zeta(3) 24 2 4 \ 2 1330 Pi 119 Pi | 1 + 280 Zeta(3) - -------- + ------- + 756 ln(n) + 756 gamma|/n + O(----) 3 8 / 2 n and in Maple notation 7/6*Pi^4*Zeta(3)+504*(ln(n)+gamma)*Zeta(5)-84*Pi^2*Zeta(5)+210*(ln(n)+gamma)^2* Zeta(3)+1540*(ln(n)+gamma)*Zeta(3)-175/2*(ln(n)+gamma)^2*Pi^2+49/12*(ln(n)+ gamma)*Pi^4-343/3*(ln(n)+gamma)*Pi^2+56*(ln(n)+gamma)^2+720*Zeta(7)-1400/3*Pi^2 *Zeta(3)+3360*Zeta(5)+105*(ln(n)+gamma)^3-5/8*Pi^6+602*Zeta(3)+840*Zeta(3)^2-21 /2*Pi^2+35/6*Pi^4-70*(ln(n)+gamma)*Pi^2*Zeta(3)+ln(n)+gamma+(210*(-ln(n)-gamma) *Zeta(3)+1960*(ln(n)+gamma)*Zeta(3)-420*(-ln(n)-gamma-1/12*Pi^2)*Zeta(3)-105/2* (ln(n)+gamma)^2*Pi^2-175/2*(-ln(n)-gamma)*Pi^2+7/4*(ln(n)+gamma)*Pi^4-630*(ln(n )+gamma)*Pi^2+1883/2*(ln(n)+gamma)^2+105*(ln(n)+gamma)*(-ln(n)-gamma)+125/2-35* (ln(n)+gamma)^4+21*(ln(n)+gamma)^5-490*Pi^2*Zeta(3)+3276*Zeta(5)+140*(ln(n)+ gamma)^3-5/24*Pi^6-7*(ln(n)+gamma)^6+4830*Zeta(3)+280*Zeta(3)^2-1330/3*Pi^2+119 /8*Pi^4+756*ln(n)+756*gamma)/n+O(1/n^2) and in floating-point 221.3365550+813.640411*ln(n)-555.1584358*(ln(n)+.5772156649)^2+105.*(ln(n)+.577\ 2156649)^3+(-1819.331002*ln(n)+94.9525557+423.3457688*(ln(n)+.5772156649)^2+105\ .*(ln(n)+.5772156649)*(-1.*ln(n)-.5772156649)-35.*(ln(n)+.5772156649)^4+21.*(ln (n)+.5772156649)^5+140.*(ln(n)+.5772156649)^3-7.*(ln(n)+.5772156649)^6)/n+O(1/n ^2) Theorem Number, 8, : The , 8, -th moment about the mean is 7 6 4 3 2 2 (8 Hn[1] - 28 Hn[1] + 105 n Hn[1] - 420 n Hn[1] Hn[2] + 630 n Hn[1] Hn[2] 3 4 5 3 - 420 n Hn[1] Hn[2] + 105 n Hn[2] + 56 Hn[1] + 490 n Hn[1] 2 2 2 - 3850 n Hn[1] Hn[2] + 3640 n Hn[1] Hn[3] - 1260 n Hn[1] Hn[4] 2 + 7350 n Hn[1] Hn[2] - 9520 n Hn[1] Hn[2] Hn[3] 2 3 + 2520 n Hn[1] Hn[2] Hn[4] + 1120 n Hn[1] Hn[3] - 3990 n Hn[2] 2 2 2 + 5880 n Hn[2] Hn[3] - 1260 n Hn[2] Hn[4] - 1120 n Hn[2] Hn[3] 4 2 - 70 Hn[1] + 119 n Hn[1] - 2394 n Hn[1] Hn[2] + 8792 n Hn[1] Hn[3] - 14700 n Hn[1] Hn[4] + 11424 n Hn[1] Hn[5] - 3360 n Hn[1] Hn[6] 2 + 5103 n Hn[2] - 21000 n Hn[2] Hn[3] + 23940 n Hn[2] Hn[4] 2 - 14112 n Hn[2] Hn[5] + 3360 n Hn[2] Hn[6] + 10640 n Hn[3] 2 3 - 11760 n Hn[3] Hn[4] + 2688 n Hn[3] Hn[5] + 1260 n Hn[4] + 1316 Hn[1] 2 2 2 - 5460 Hn[1] Hn[2] + 1680 Hn[1] Hn[3] + 7140 Hn[1] Hn[2] 3 2 - 3360 Hn[1] Hn[2] Hn[3] - 2940 Hn[2] + 1680 Hn[2] Hn[3] + n Hn[1] - 127 n Hn[2] + 1932 n Hn[3] - 10206 n Hn[4] + 25200 n Hn[5] 2 - 31920 n Hn[6] + 20160 n Hn[7] - 5040 n Hn[8] + 1890 Hn[1] - 13188 Hn[1] Hn[2] + 19600 Hn[1] Hn[3] - 14280 Hn[1] Hn[4] 2 + 4032 Hn[1] Hn[5] + 15750 Hn[2] - 31920 Hn[2] Hn[3] + 17640 Hn[2] Hn[4] 2 - 4032 Hn[2] Hn[5] + 7840 Hn[3] - 3360 Hn[3] Hn[4] + 254 Hn[1] - 2898 Hn[2] + 13608 Hn[3] - 31500 Hn[4] + 38304 Hn[5] - 23520 Hn[6] + 5760 Hn[7])/n and in Maple notation (8*Hn[1]^7-28*Hn[1]^6+105*n*Hn[1]^4-420*n*Hn[1]^3*Hn[2]+630*n*Hn[1]^2*Hn[2]^2-\ 420*n*Hn[1]*Hn[2]^3+105*n*Hn[2]^4+56*Hn[1]^5+490*n*Hn[1]^3-3850*n*Hn[1]^2*Hn[2] +3640*n*Hn[1]^2*Hn[3]-1260*n*Hn[1]^2*Hn[4]+7350*n*Hn[1]*Hn[2]^2-9520*n*Hn[1]*Hn [2]*Hn[3]+2520*n*Hn[1]*Hn[2]*Hn[4]+1120*n*Hn[1]*Hn[3]^2-3990*n*Hn[2]^3+5880*n* Hn[2]^2*Hn[3]-1260*n*Hn[2]^2*Hn[4]-1120*n*Hn[2]*Hn[3]^2-70*Hn[1]^4+119*n*Hn[1]^ 2-2394*n*Hn[1]*Hn[2]+8792*n*Hn[1]*Hn[3]-14700*n*Hn[1]*Hn[4]+11424*n*Hn[1]*Hn[5] -3360*n*Hn[1]*Hn[6]+5103*n*Hn[2]^2-21000*n*Hn[2]*Hn[3]+23940*n*Hn[2]*Hn[4]-\ 14112*n*Hn[2]*Hn[5]+3360*n*Hn[2]*Hn[6]+10640*n*Hn[3]^2-11760*n*Hn[3]*Hn[4]+2688 *n*Hn[3]*Hn[5]+1260*n*Hn[4]^2+1316*Hn[1]^3-5460*Hn[1]^2*Hn[2]+1680*Hn[1]^2*Hn[3 ]+7140*Hn[1]*Hn[2]^2-3360*Hn[1]*Hn[2]*Hn[3]-2940*Hn[2]^3+1680*Hn[2]^2*Hn[3]+n* Hn[1]-127*n*Hn[2]+1932*n*Hn[3]-10206*n*Hn[4]+25200*n*Hn[5]-31920*n*Hn[6]+20160* n*Hn[7]-5040*n*Hn[8]+1890*Hn[1]^2-13188*Hn[1]*Hn[2]+19600*Hn[1]*Hn[3]-14280*Hn[ 1]*Hn[4]+4032*Hn[1]*Hn[5]+15750*Hn[2]^2-31920*Hn[2]*Hn[3]+17640*Hn[2]*Hn[4]-\ 4032*Hn[2]*Hn[5]+7840*Hn[3]^2-3360*Hn[3]*Hn[4]+254*Hn[1]-2898*Hn[2]+13608*Hn[3] -31500*Hn[4]+38304*Hn[5]-23520*Hn[6]+5760*Hn[7])/n Corollary Number, 8, : The asymptotics of the variance to order, 2, is 2 11424 (ln(n) + gamma) Zeta(5) + 3640 (ln(n) + gamma) Zeta(3) 4 2 2 2 + 98/3 Pi Zeta(3) - 2352 Pi Zeta(5) - 1925/3 (ln(n) + gamma) Pi 4 2 + 245/6 (ln(n) + gamma) Pi + 20160 Zeta(7) - 3500 Pi Zeta(3) 3 2 2 4 + 25200 Zeta(5) - 70 (ln(n) + gamma) Pi + 7/2 (ln(n) + gamma) Pi / 6 2 | - 5/6 (ln(n) + gamma) Pi + 1120 (ln(n) + gamma) Zeta(3) + | \ 2 4032 (ln(n) + gamma) Zeta(5) + 1680 (ln(n) + gamma) Zeta(3) / 2\ | Pi | + 3640 (-ln(n) - gamma) Zeta(3) - 9520 |-ln(n) - gamma - ---| Zeta(3) \ 12 / 4 2 2 2 + 28/3 Pi Zeta(3) - 672 Pi Zeta(5) - 1120 (ln(n) + gamma) Pi 4 + 224/3 (ln(n) + gamma) Pi + 490 (ln(n) + gamma) (-ln(n) - gamma) 2 2 - 1925/3 (-ln(n) - gamma) Pi + 5760 Zeta(7) - 7280 Pi Zeta(3) / 2\ | Pi | 4 + 46704 Zeta(5) + 28 |-ln(n) - gamma - ---| Pi \ 12 / 2 2 - 70 ((ln(n) + gamma) (-ln(n) - gamma) - 1/2 (ln(n) + gamma) ) Pi 4 2 + 7/2 (-ln(n) - gamma) Pi + 210 (ln(n) + gamma) (-ln(n) - gamma) 6 47 Pi 2 2 - ------ + 5495 (ln(n) + gamma) + 30212 Zeta(3) + 8400 Zeta(3) 12 3 2 5 + 1736 (ln(n) + gamma) - 4648 (ln(n) + gamma) Pi + 56 (ln(n) + gamma) 2 4 3969 Pi 1603 Pi 2 - -------- + -------- - 560 (ln(n) + gamma) Pi Zeta(3) + 2529 ln(n) 2 12 6 7 - 28 (ln(n) + gamma) + 19600 (ln(n) + gamma) Zeta(3) + 8 (ln(n) + gamma) \ 4 | + 2529 gamma - 70 (ln(n) + gamma) + 253/2|/n + 2688 Zeta(3) Zeta(5) / 6 2 2 95 Pi 2 - 560/3 Pi Zeta(3) - ------ + 119 (ln(n) + gamma) + 1932 Zeta(3) 12 8 2 3 67 Pi 2 + 10640 Zeta(3) + 490 (ln(n) + gamma) - ------ - 399 (ln(n) + gamma) Pi 720 2 4 127 Pi 567 Pi 2 1 - ------- + ------- - 4760/3 (ln(n) + gamma) Pi Zeta(3) + ln(n) + O(----) 6 20 2 n 4 + 8792 (ln(n) + gamma) Zeta(3) + gamma + 105 (ln(n) + gamma) and in Maple notation 11424*(ln(n)+gamma)*Zeta(5)+3640*(ln(n)+gamma)^2*Zeta(3)+98/3*Pi^4*Zeta(3)-2352 *Pi^2*Zeta(5)-1925/3*(ln(n)+gamma)^2*Pi^2+245/6*(ln(n)+gamma)*Pi^4+20160*Zeta(7 )-3500*Pi^2*Zeta(3)+25200*Zeta(5)-70*(ln(n)+gamma)^3*Pi^2+7/2*(ln(n)+gamma)^2* Pi^4-5/6*(ln(n)+gamma)*Pi^6+1120*(ln(n)+gamma)*Zeta(3)^2+(4032*(ln(n)+gamma)* Zeta(5)+1680*(ln(n)+gamma)^2*Zeta(3)+3640*(-ln(n)-gamma)*Zeta(3)-9520*(-ln(n)- gamma-1/12*Pi^2)*Zeta(3)+28/3*Pi^4*Zeta(3)-672*Pi^2*Zeta(5)-1120*(ln(n)+gamma)^ 2*Pi^2+224/3*(ln(n)+gamma)*Pi^4+490*(ln(n)+gamma)*(-ln(n)-gamma)-1925/3*(-ln(n) -gamma)*Pi^2+5760*Zeta(7)-7280*Pi^2*Zeta(3)+46704*Zeta(5)+28*(-ln(n)-gamma-1/12 *Pi^2)*Pi^4-70*((ln(n)+gamma)*(-ln(n)-gamma)-1/2*(ln(n)+gamma)^2)*Pi^2+7/2*(-ln (n)-gamma)*Pi^4+210*(ln(n)+gamma)^2*(-ln(n)-gamma)-47/12*Pi^6+5495*(ln(n)+gamma )^2+30212*Zeta(3)+8400*Zeta(3)^2+1736*(ln(n)+gamma)^3-4648*(ln(n)+gamma)*Pi^2+ 56*(ln(n)+gamma)^5-3969/2*Pi^2+1603/12*Pi^4-560*(ln(n)+gamma)*Pi^2*Zeta(3)+2529 *ln(n)-28*(ln(n)+gamma)^6+19600*(ln(n)+gamma)*Zeta(3)+8*(ln(n)+gamma)^7+2529* gamma-70*(ln(n)+gamma)^4+253/2)/n+2688*Zeta(3)*Zeta(5)-560/3*Pi^2*Zeta(3)^2-95/ 12*Pi^6+119*(ln(n)+gamma)^2+1932*Zeta(3)+10640*Zeta(3)^2+490*(ln(n)+gamma)^3-67 /720*Pi^8-399*(ln(n)+gamma)*Pi^2-127/6*Pi^2+567/20*Pi^4-4760/3*(ln(n)+gamma)*Pi ^2*Zeta(3)+ln(n)+O(1/n^2)+8792*(ln(n)+gamma)*Zeta(3)+gamma+105*(ln(n)+gamma)^4 and in floating-point 4448.151300*ln(n)-298.8694653-1497.577214*(ln(n)+.5772156649)^2-200.8723083*(ln (n)+.5772156649)^3+(-4641.539367*ln(n)+4814.334707-3194.065179*(ln(n)+.57721566\ 49)^2-200.8723083*(ln(n)+.5772156649)*(-1.*ln(n)-.5772156649)+210.*(ln(n)+.5772\ 156649)^2*(-1.*ln(n)-.5772156649)+1736.*(ln(n)+.5772156649)^3+56.*(ln(n)+.57721\ 56649)^5-28.*(ln(n)+.5772156649)^6+8.*(ln(n)+.5772156649)^7-70.*(ln(n)+.5772156\ 649)^4)/n+O(1/n^2)+105.*(ln(n)+.5772156649)^4 -----------------------