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

0,2

Links

Table of n, a(n) for n=0..14.

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.

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.

Example

-0.089326522343551...

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]

Crossrefs

Cf. A074760, A104539, A104540, A242056.

Sequence in context: A339800 A197392 A021922 * A257438 A197826 A197755

Adjacent sequences: A246840 A246841 A246842 * A246844 A246845 A246846

Keyword

nonn,cons,more

Author

Jean-François Alcover, Sep 05 2014

Extensions

Typo in the formula corrected by Vaclav Kotesovec, Sep 17 2014

Status

approved