A246843 Decimal expansion of C, a constant associated with the estimation of the maximum of |zeta(1+i*t)|.
0, 8, 9, 3, 2, 6, 5, 2, 2, 3, 4, 3, 5, 5, 1
Offset: 0
Examples
-0.089326522343551...
Links
- Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 28.
- Tadej Kotnik, Computational estimation of the constant β(1) characterizing the order of ζ(1 + it), Math. Comp. 77: 1713-1723, 2008.
Programs
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Mathematica
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]
Formula
1 - log(2) + integral_{0..2} log(BesselI(0, t))/t^2 dt + integral_{2..infinity} (log(BesselI(0, t)) - t)/t^2 dt.
Extensions
Typo in the formula corrected by Vaclav Kotesovec, Sep 17 2014