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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. 2
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

LINKS

Table of n, a(n) for n=0..106.

FORMULA

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

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

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

EXAMPLE

0.9271841545163232751394136...

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]]

CROSSREFS

Cf. A001620, A073002, A082633.

Sequence in context: A179638 A224268 A019877 * A336912 A010537 A234371

Adjacent sequences:  A252895 A252896 A252897 * A252899 A252900 A252901

KEYWORD

nonn,cons

AUTHOR

Richard R. Forberg, Dec 24 2014

EXTENSIONS

More digits from Alois P. Heinz, Dec 30 2014

STATUS

approved

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Last modified April 21 16:50 EDT 2021. Contains 343156 sequences. (Running on oeis4.)