%I #6 Jul 22 2022 04:31:33

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

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

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

%N Decimal expansion of 1 + log(2*Pi) - 2*gamma, where gamma is Euler's constant (A001620).

%C The constant c in the asymptotic formula for the second moment of the Riemann zeta function on the critical line Re(z) = 1/2: Integral_{t=0..T} |zeta(1/2 + i*t)|^2 dt ~ (log(T) - c) * T.

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

%H F. V. Atkinson, <a href="https://doi.org/10.1093/qmath/os-10.1.122">The mean value of the zeta-function on the critical line</a>, The Quarterly Journal of Mathematics, Vol. os-10, No. 1 (1939), pp. 122-128.

%H A. E. Ingham, <a href="https://doi.org/10.1112/plms/s2-27.1.273">Mean-value theorems in the theory of the Riemann zeta-function</a>, Proceedings of the London Mathematical Society, Vol. s2-27, No. 1 (1928), pp. 273-300.

%H E. C. Titchmarsh, <a href="https://doi.org/10.1093/qmath/os-5.1.195">On van der Corput's method and the zeta-function of Riemann (V)</a>, The Quarterly Journal of Mathematics, Vol. os-5, No. 1 (1934), pp. 195-210.

%e 1.68344573660627976234763529264643041763847627539571...

%t RealDigits[1 + Log[2*Pi] - 2*EulerGamma, 10, 100][[1]]

%Y Cf. A001620, A053510, A355977.

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, Jul 22 2022