OFFSET
1,1
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
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)]
MATHEMATICA
RealDigits[1/2+2/Sqrt[3]+2/Sqrt[5], 10, 120][[1]] (* Harvey P. Dale, Jul 31 2013 *)
PROG
(PARI) 1/2 + 2/sqrt(3) + 2/sqrt(5) \\ Charles R Greathouse IV, Mar 10 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Mats Granvik, Jul 20 2012
EXTENSIONS
Corrected by Harvey P. Dale, Jul 31 2013
STATUS
approved