A271798 Decimal expansion of Matthews' constant C_2, an analog of Artin's constant for primitive roots.

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

Offset: 0

Author

Jean-François Alcover, Apr 14 2016

Keywords

• nonn
• cons

Examples

			0.147349400002001458076808431847692596396671858173272158442...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.

Links

• K. R. Matthews, A generalisation of Artin's conjecture for primitive roots, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.

Crossrefs

Cf. A005596.

Programs

Mathematica

digits = 66; m0 = 1000; dm = 100; Clear[s]; r[n_] := RootSum[1 - 2*# - #^2 + #^3& , #^n&] - 1; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m - dm], 10, digits][[1]], Print[m]; m = m + dm]; RealDigits[s[m]][[1]]

PARI

prodeulerrat(1 - (p^2 - (p - 1)^2)/(p^2*(p - 1))) \\ Amiram Eldar, Mar 16 2021

Formula

C_2 = Product_{p prime} 1 - (p^2 - (p - 1)^2)/(p^2*(p - 1)).

Log(1 - (p^2 - (p - 1)^2)/(p^2*(p - 1))) + O(p,Infinity)^m = Sum_{n=2..m} -r(n)/(n*p^n), where r(n) = rootSum(1 - 2*x - x^2 + x^3, x^n) - 1.

Extensions

More digits from Vaclav Kotesovec, Jun 19 2020