A252898 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.

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

Offset: 0

Author

Richard R. Forberg, Dec 24 2014

Keywords

• nonn
• cons

Comments

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.

Examples

			0.9271841545163232751394136...
		

Crossrefs

Cf. A001620, A073002, A082633.

Programs

Maple

s:= convert(evalf(1+gamma(1), 140), string):
seq(parse(s[n+2]), n=0..110);  # Alois P. Heinz, Dec 30 2014

Mathematica

FractionalPart[N[Derivative[1][Zeta][
   1 + 1/(1000000000000000000000000000000000000000000000000000000000000)], 400]]

Formula

Limit_{n -> infinity} -FractionalPart[Zeta'(1+1/n)]

Limit_{n -> infinity} -FractionalPart[Zeta'(1-1/n)]

Equals 1 - A082633. - Alois P. Heinz, Dec 30 2014

Extensions

More digits from Alois P. Heinz, Dec 30 2014