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Decimal expansion of Sum{p prime>=2} log(p)/(p^2-p+1).
8

%I #33 Sep 18 2022 14:31:15

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

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

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

%N Decimal expansion of Sum{p prime>=2} log(p)/(p^2-p+1).

%C Appears in the asymptotic formula for Sum{k=1..n} 1/phi(k), with phi(k) being Euler's totient function. - _Stanislav Sykora_, Nov 14 2014

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.7 Euler totient constants, p. 116.

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant Z2).

%H X. Gourdon, P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's totient function</a>

%F Equals lim_{n->infinity} (Gamma + log(n) - c*Sum_{k=1..n} 1/phi(k)), where Gamma is the Euler-Mascheroni constant, and c = zeta(6)/(zeta(2)*zeta(3)) = 1/A082695. This equals further lim_{n->infinity} Sum{k=1..n} (1/k - c/phi(k)) and lim_{n->infinity}(A001008(n)/A002805(n) - (A028415(n)/A048049(n))/A082695). - _Stanislav Sykora_, Nov 15 2014

%e 0.60838171786332472...

%t digits = 105; m0 = 100; dm = 100; Clear[s]; s[n_] := s[n] = Sum[ Switch[ Mod[k, 6], 0, 1, 1, 0, 2, -1, 3, -1, 4, 0, 5, 1] * PrimeZetaP'[k], {k, 2, n}] // N[#, digits+40]&; Print[m0, " ", s[m0]]; s[m = m0+dm]; While[ Print[m, " ", s[m]]; RealDigits[s[m], 10, digits+5] != RealDigits[s[m-dm], 10, digits+5], m = m+dm]; RealDigits[s[m], 10, digits] // First (* _Jean-François Alcover_, Sep 11 2015 *)

%Y Cf. A001008, A002805, A028415, A048049, A082695.

%K cons,nonn

%O 0,1

%A Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 07 2003

%E More terms from _Benoit Cloitre_, Mar 06 2013

%E More digits from _Jean-François Alcover_, Sep 11 2015