A355977 - OEIS

A355977

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

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, 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, 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

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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

Table of n, a(n) for n=2..106.

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]]