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%I #38 Jan 09 2024 12:34:21
%S 9,2,7,1,8,4,1,5,4,5,1,6,3,2,3,2,7,5,1,3,9,4,1,3,6,2,4,1,2,5,0,9,8,6,
%T 8,0,8,6,2,2,6,3,6,6,1,6,6,5,6,6,2,0,4,7,4,0,0,9,9,3,4,4,0,2,5,8,5,9,
%U 8,5,6,6,4,2,8,4,8,8,5,1,5,1,2,1,9,1,3,0,7,1,7,5,5,1,5,5,9,8,5,3,9,5,9,2,2,7,9
%N Decimal expansion of lim_{n->infinity} -FractionalPart[Zeta'(1+1/n)] or -FractionalPart[Zeta'(1-1/n)], where Zeta' is the first derivative of the Riemann zeta function.
%C Zeta'(x) -> negative infinity as x -> 1, from above and below.
%C When 1 is approached using arguments of (1+1/n) or (1-1/n), its fractional part converges to this constant.
%C The Euler-Mascheroni constant is the fractional part as x->1 for Zeta(x), but with a different symmetry approaching 1 from above vs. below. See A001620 and below.
%C The integer part of Zeta'(1 + 1/n) or Zeta'(1 - 1/n) = -(n^2 - 1).
%C Corresponding constants, as taken from the fractional part, exist for the higher order derivatives of the Riemann Zeta as x->1 with these arguments. The list below shows converged values up to the 10th derivative approaching 1 from above, using
%C x = 1 + 1/n, as n -> infinity, with signs:
%C Derivative[1] = -0.9271841545163232751394136... (this entry)
%C Derivative[2] = 0.9903096368071276815154696...
%C Derivative[3] = -0.0020538344203033458661600...
%C Derivative[4] = 0.0023253700654673000057468...
%C Derivative[5] = -0.0007933238173010627017533...
%C Derivative[6] = 0.9997612306545698003901275...
%C Derivative[7] = -0.9994727104329422489539259...
%C Derivative[8] = 0.9996478766461969604903979...
%C Derivative[9] = -0.9999656052255819119518220...
%C Derivative[10]= 0.0002053328149090647946837...
%C Even order derivatives, D[2m], (e.g., 2nd, 4th, 6th, ...) have different fractional values when approaching 1 from below equal to: -(1-D[2m]). The same is true for D[0], or Zeta itself.
%C The integer sequences associated with the integer part, with x ->1 from above and starting with the argument x= 2 = 1+1/n, hence n = 1 to infinity, are:
%C Derivative[1] = -(n^2-1)
%C Derivative[2] = (2!*n^3-1)
%C Derivative[3] = -(3!*n^4)
%C Derivative[4] = (4!*n^5)
%C Derivative[5] = -(5!*n^6)
%C Derivative[6] = (6!*n^7-1), except at n=1, where value = 720 with fract ~0.0001
%C Derivative[7] = -(7!*n^8-1)
%C Derivative[8] = (8!*n^9-1)
%C Derivative[9] = -(9!*n^10-1)
%C Derivative[10] = (10!*n^11)
%C Thus, rounding the m-th derivative of Zeta(x) at x=2 (n=1) gives (-1)^m * m! for m>=1. See A073002.
%F Limit_{n -> infinity} -FractionalPart[Zeta'(1+1/n)]
%F Limit_{n -> infinity} -FractionalPart[Zeta'(1-1/n)]
%F Equals 1-A082633. - _Alois P. Heinz_, Dec 30 2014
%e 0.9271841545163232751394136...
%p s:= convert(evalf(1+gamma(1), 140), string):
%p seq(parse(s[n+2]), n=0..110); # _Alois P. Heinz_, Dec 30 2014
%t FractionalPart[N[Derivative[1][Zeta][
%t 1 + 1/(1000000000000000000000000000000000000000000000000000000000000)], 400]]
%Y Cf. A001620, A073002, A082633.
%K nonn,cons
%O 0,1
%A _Richard R. Forberg_, Dec 24 2014
%E More digits from _Alois P. Heinz_, Dec 30 2014
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