A214533 Decimal expansion of 1/2 + 2/sqrt(3) + 2/sqrt(5).

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

Offset: 1

Author

Mats Granvik, Jul 20 2012

Keywords

• nonn
• cons

Comments

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.

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

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.

ZetaZero[18]/2/2.549127729379167407581

ZetaZero[1]

14.135650568603255663

14.134725141734693790

ZetaZero[33]/2/2.549127729379167407581

ZetaZero[2]

21.020643640006420723

21.022039638771554993

ZetaZero[42]/2/2.549127729379167407581

ZetaZero[3]

25.011827067342131577

25.010857580145688763

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:

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, ...

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

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

Links

• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A191898.

Programs

Mathematica

RealDigits[1/2+2/Sqrt[3]+2/Sqrt[5],10,120][[1]] (* Harvey P. Dale, Jul 31 2013 *)

PARI

1/2 + 2/sqrt(3) + 2/sqrt(5) \\ Charles R Greathouse IV, Mar 10 2016

Formula

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)]

Extensions

Corrected by Harvey P. Dale, Jul 31 2013