OFFSET
2,3
COMMENTS
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.
REFERENCES
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 177.
LINKS
D. R. Heath-Brown, The mean value theorem for the Riemann zeta-function, Mathematika, Vol. 25, No. 2 (1978), pp. 177-184.
Tom Meurman, On the mean square of the Riemann zeta-function, The Quarterly Journal of Mathematics, Vol. 38, No. 3 (1987), pp. 337-343.
FORMULA
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.
EXAMPLE
10.30471743950013870996889214117176357043735998020794...
MATHEMATICA
RealDigits[2*Zeta[3/2]^4/(3*Sqrt[2*Pi]*Zeta[3]), 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jul 22 2022
STATUS
approved