A355977 - OEIS

%I #8 Jan 08 2025 11:36:36

%S 1,0,3,0,4,7,1,7,4,3,9,5,0,0,1,3,8,7,0,9,9,6,8,8,9,2,1,4,1,1,7,1,7,6,

%T 3,5,7,0,4,3,7,3,5,9,9,8,0,2,0,7,9,4,6,6,2,3,8,5,0,4,5,3,5,9,2,8,5,1,

%U 6,6,8,4,5,2,7,4,3,8,0,3,2,8,6,7,8,5,4,5,3,3,2,6,9,8,5,4,9,0,3,0,8,0,6,8,1

%N Decimal expansion of 2*zeta(3/2)^4/(3*sqrt(2*Pi)*zeta(3)).

%C The constant c_2 in the asymptotic mean of the squared error of the second moment of the Riemann zeta function on the critical line Re(z) = 1/2: Integral_{t=2..T} E(t)^2 dt ~ c_2 * T^(3/2), where E(t) = Integral_{t=0..T} |zeta(1/2 + i*t)|^2 dt - (log(T) - c) * T, and c is A355976.

%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 177.

%H D. R. Heath-Brown, <a href="https://doi.org/10.1112/S0025579300009414">The mean value theorem for the Riemann zeta-function</a>, Mathematika, Vol. 25, No. 2 (1978), pp. 177-184.

%H Tom Meurman, <a href="https://doi.org/10.1093/qmath/38.3.337">On the mean square of the Riemann zeta-function</a>, The Quarterly Journal of Mathematics, Vol. 38, No. 3 (1987), pp. 337-343.

%F Equals (2/(3*sqrt(2*Pi)) * Sum_{k>=1} d(k)^2/k^(3/2), where d(k) = A000005(k) is the number of divisors of k.

%e 10.30471743950013870996889214117176357043735998020794...

%t RealDigits[2*Zeta[3/2]^4/(3*Sqrt[2*Pi]*Zeta[3]), 10, 100][[1]]

%Y Cf. A000005, A002117, A019727, A078434, A355976.

%K nonn,cons

%O 2,3

%A _Amiram Eldar_, Jul 22 2022