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.
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, 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, 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
Offset: 1
Examples
2.176968743559410321739727298735814329767273758965844960238628...
Links
- Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 16.
- Kevin Ford, The distribution of Totients
Programs
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Mathematica
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
Formula
Let F(x) = sum_{k >= 1} ((k+1)*log(k+1) - k*log(k) - 1)*x^k.
C = 1/(2*|log(rho)|), where rho is the unique solution on [0,1) of F(rho)=1.
D = 2*C*(1 + log(F'(rho)) - log(2*C)) - 3/2.