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Decimal expansion of 1/2 + 2/sqrt(3) + 2/sqrt(5).
0

%I #20 Aug 21 2023 11:38:22

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

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

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

%N Decimal expansion of 1/2 + 2/sqrt(3) + 2/sqrt(5).

%C The convergent of a sum of reciprocals of square roots with numerators equal to the numerators in the Dirichlet series for Mangoldt Lambda [6] = 0.

%C Superposition of Dirichlet series of 6 shifted versions of A100051 evaluated at s=1/2.

%C The nontrivial Riemann zeta zeros are known to not be multiples of any number. This number -2.5491277293... comes close to relating the 18th, 33rd and 42nd zeta zeros to the first, second, and third zeta zeros, respectively.

%C ZetaZero[18]/2/2.549127729379167407581

%C ZetaZero[1]

%C 14.135650568603255663

%C 14.134725141734693790

%C ZetaZero[33]/2/2.549127729379167407581

%C ZetaZero[2]

%C 21.020643640006420723

%C 21.022039638771554993

%C ZetaZero[42]/2/2.549127729379167407581

%C ZetaZero[3]

%C 25.011827067342131577

%C 25.010857580145688763

%C Numerators in the sum for this constant are the sixth row and column in matrix A191898. The increment in the denominators is equal to 1, and the denominators begin:

%C 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 9, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, ...

%C Sums of this type that have numerators equal to Dirichlet series for logarithms are partials sums of square roots.

%C An algebraic number of degree 4 and denominator 30; minimal polynomial 3600x^4 - 7200x^3 - 9960x^2 + 13560x - 2591. - _Charles R Greathouse IV_, Apr 21 2016

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>

%F Equals the absolute value of sum_{n=1..infinity} [1/(n + 0)^(1/2) - 1/(n + 1)^(1/2) - 2/(n + 2)^(1/2) - 1/(n + 3)^(1/2) + 1/(n + 4)^(1/2) + 2/(n + 5)^(1/2)]

%t RealDigits[1/2+2/Sqrt[3]+2/Sqrt[5],10,120][[1]] (* _Harvey P. Dale_, Jul 31 2013 *)

%o (PARI) 1/2 + 2/sqrt(3) + 2/sqrt(5) \\ _Charles R Greathouse IV_, Mar 10 2016

%Y Cf. A191898.

%K nonn,cons

%O 1,1

%A _Mats Granvik_, Jul 20 2012

%E Corrected by _Harvey P. Dale_, Jul 31 2013