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A252898
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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.
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2
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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, 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, 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
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OFFSET
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0,1
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COMMENTS
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Zeta'(x) -> negative infinity as x -> 1, from above and below.
When 1 is approached using arguments of (1+1/n) or (1-1/n), its fractional part converges to this constant.
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.
The integer part of Zeta'(1 + 1/n) or Zeta'(1 - 1/n) = -(n^2 - 1).
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
x = 1 + 1/n, as n -> infinity, with signs:
Derivative[1] = -0.9271841545163232751394136... (this entry)
Derivative[2] = 0.9903096368071276815154696...
Derivative[3] = -0.0020538344203033458661600...
Derivative[4] = 0.0023253700654673000057468...
Derivative[5] = -0.0007933238173010627017533...
Derivative[6] = 0.9997612306545698003901275...
Derivative[7] = -0.9994727104329422489539259...
Derivative[8] = 0.9996478766461969604903979...
Derivative[9] = -0.9999656052255819119518220...
Derivative[10]= 0.0002053328149090647946837...
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.
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:
Derivative[1] = -(n^2-1)
Derivative[2] = (2!*n^3-1)
Derivative[3] = -(3!*n^4)
Derivative[4] = (4!*n^5)
Derivative[5] = -(5!*n^6)
Derivative[6] = (6!*n^7-1), except at n=1, where value = 720 with fract ~0.0001
Derivative[7] = -(7!*n^8-1)
Derivative[8] = (8!*n^9-1)
Derivative[9] = -(9!*n^10-1)
Derivative[10] = (10!*n^11)
Thus, rounding the m-th derivative of Zeta(x) at x=2 (n=1) gives (-1)^m * m! for m>=1. See A073002.
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LINKS
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FORMULA
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Limit_{n -> infinity} -FractionalPart[Zeta'(1+1/n)]
Limit_{n -> infinity} -FractionalPart[Zeta'(1-1/n)]
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EXAMPLE
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0.9271841545163232751394136...
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MAPLE
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s:= convert(evalf(1+gamma(1), 140), string):
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MATHEMATICA
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FractionalPart[N[Derivative[1][Zeta][
1 + 1/(1000000000000000000000000000000000000000000000000000000000000)], 400]]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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