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%I #16 Jan 17 2020 16:09:30
%S 0,8,9,3,2,6,5,2,2,3,4,3,5,5,1
%N Decimal expansion of C, a constant associated with the estimation of the maximum of |zeta(1+i*t)|.
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 28.
%H Tadej Kotnik, <a href="http://dx.doi.org/10.1090/S0025-5718-08-02065-6">Computational estimation of the constant β(1) characterizing the order of ζ(1 + it)</a>, Math. Comp. 77: 1713-1723, 2008.
%F 1 - log(2) + integral_{0..2} log(BesselI(0, t))/t^2 dt + integral_{2..infinity} (log(BesselI(0, t)) - t)/t^2 dt.
%e -0.089326522343551...
%t digits = 15; precision = 200; u0 = 10^8; du = 10^8; tail[u_] := -(1 + Log[2*Pi*u])/(2*u); Clear[f]; f[u_] := f[u] = 1 - Log[2] + NIntegrate[Log[BesselI[0, t]]/t^2, {t, 0, 2} , WorkingPrecision -> precision] + NIntegrate[(Log[BesselI[0, t]] - t)/t^2, {t, 2, u}, WorkingPrecision -> precision, MaxRecursion -> 20 ] + tail[u]; f[u0]; f[u = u0 + du]; While[RealDigits[f[u], 10, digits + 4] != RealDigits[f[u - du], 10, digits + 4], Print["u = ", u, " ", f[u]]; u = u + du]; Join[{0}, RealDigits[f[u], 10, digits] // First]
%Y Cf. A074760, A104539, A104540, A242056.
%K nonn,cons,more
%O 0,2
%A _Jean-François Alcover_, Sep 05 2014
%E Typo in the formula corrected by _Vaclav Kotesovec_, Sep 17 2014