cp's OEIS Frontend

A246751 Decimal expansion of D, an auxiliary constant associated with the asymptotic number of values of the Euler totient function less than a given number.

%I A246751 #12 May 18 2020 10:33:23
%S A246751 2,1,7,6,9,6,8,7,4,3,5,5,9,4,1,0,3,2,1,7,3,9,7,2,7,2,9,8,7,3,5,8,1,4,
%T A246751 3,2,9,7,6,7,2,7,3,7,5,8,9,6,5,8,4,4,9,6,0,2,3,8,6,2,8,0,0,0,6,4,7,3,
%U A246751 5,2,5,6,2,2,0,3,3,7,4,9,0,9,8,4,0,5,1,2,2,7,4,0,8,6,0,7,4,9,3
%N A246751 Decimal expansion of D, an auxiliary constant associated with the asymptotic number of values of the Euler totient function less than a given number.
%H A246751 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 16.
%H A246751 Kevin Ford, <a href="http://www.math.uiuc.edu/~ford/wwwpapers/totients.pdf">The distribution of Totients</a>
%F A246751 Let F(x) = sum_{k >= 1} ((k+1)*log(k+1) - k*log(k) - 1)*x^k.
%F A246751 C = 1/(2*|log(rho)|), where rho is the unique solution on [0,1) of F(rho)=1.
%F A246751 D = 2*C*(1 + log(F'(rho)) - log(2*C)) - 3/2.
%e A246751 2.176968743559410321739727298735814329767273758965844960238628...
%t A246751 digits = 99; F[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*x^k, {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; F'[x_?NumericQ] := NSum[((k + 1)*Log[k + 1] - k*Log[k] - 1)*k*x^(k - 1), {k, 1, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 1000]; rho = x /. FindRoot[F[x] == 1, {x, 5/10, 6/10}, WorkingPrecision -> digits + 10]; c = -1/(2*Log[rho]); d = 2*c*(1 + Log[F'[rho]] - Log[2*c]) - 3/2; RealDigits[d, 10, digits] // First
%Y A246751 Cf. A246746, A246749, A234614.
%K A246751 nonn,cons
%O A246751 1,1
%A A246751 _Jean-François Alcover_, Sep 02 2014