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A221710 Decimal expansion of the integral of the logarithm of the Riemann zeta function from 2 to infinity. 1

%I #19 Feb 01 2020 14:34:47

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

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

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

%N Decimal expansion of the integral of the logarithm of the Riemann zeta function from 2 to infinity.

%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High-precision calculation of Hardy-Littlewood constants</a>, 1998, see I(2).

%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]

%H R. J. Mathar, <a href="http://arxiv.org/abs/0811.4739">Twenty digits of some integrals of the Prime Zeta Function</a>, arXiv:0811.4739, Section 2.4.

%e Integral_{s=2..infinity} log zeta(s) ds = 0.536526945921177100961719019548794400667....

%t NIntegrate[ Log[ Zeta[s]], {s, 2, Infinity}, WorkingPrecision -> 110] // RealDigits[#, 10, 100]& // First (* _Jean-François Alcover_, Jun 10 2013 *)

%o (PARI) intnum(s=2,[oo,log(2)],log(zeta(s))) \\ _Charles R Greathouse IV_, Dec 12 2013

%Y Cf. A188157.

%K nonn,cons

%O 0,1

%A _N. J. A. Sloane_, Jan 26 2013

%E More digits from _Jean-François Alcover_, Jun 10 2013

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)