A367309 -id:A367309 - cp's OEIS frontend

A367310 Continued fraction of the constant in A367309 (the area under the curve (1 - 2^(1-x)) zeta(x) from 0 to 1).

0, 1, 1, 1, 1, 18, 2, 1, 37, 3, 1, 2, 20, 1, 3, 3, 1, 1, 1, 1, 2, 1, 26, 1, 1, 1, 2, 4, 3, 1, 76, 1, 3, 1, 1, 2, 1, 1, 3, 5, 3, 1, 10, 2, 5, 1, 3, 8, 1, 3, 2, 2, 5, 2, 1, 1, 1, 6, 2, 36, 5, 1, 2, 1, 1, 1, 9, 4, 3, 1, 1, 4, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 29, 1

Offset: 0

Alejandro Malla, Nov 13 2023

Cf. A367309, A367311.

ContinuedFraction[NIntegrate[(1 - 2^(1-x)) Zeta[x], {x,0,1}, WorkingPrecision ->300]]

localprec(100); contfrac(intnum(x=0, 1, (1-2^(1-x))*zeta(x))) \\ Michel Marcus, Nov 14 2023

A367311 Maximum curvature of the curve (1 - 2^(1-x)) zeta(x) from 0 to 1.

Original entry on oeis.org

0, 6, 4, 1, 3, 9, 2, 8, 2, 0, 6, 4, 2, 5, 7, 1, 6, 8, 4, 2, 2, 0, 8, 8, 7, 1, 6, 5, 1, 2, 7, 1, 8, 1, 6, 8, 7, 3, 9, 3, 6, 5, 6, 8, 2, 8, 4, 4, 6, 4, 6, 4, 0, 1, 3, 9, 5, 5, 9, 5, 7, 7, 0, 0, 2, 2, 5, 2, 5, 7, 6, 2, 7, 9, 8, 3, 6, 9, 3, 2, 1, 7, 2, 4, 9, 4, 7

Offset: 0

Clark Kimberling and Peter J. C. Moses, Nov 13 2023

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to (1 - 2^(1-x)) zeta(x) (0,1). This series can be described as an alternating version of the "p-series" when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x and g(x) = (1 - 2^(1-x)) zeta(x). Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined. Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

			Maximum curvature = 0.0641392820642571684220887165127181687393..., which occurs at x = 0.6827548440370203586269... .

Cf. A113024, A367309, A367312.

f[x_] := (1 - 2^(1 - x)) Zeta[x];
c[x_] := Abs[f''[x]]/(1 + f'[x]^2)^(3/2)
y = FindMaximum[{c[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
RealDigits[y][[1]][[1]]

One initial 0 inserted by Artur Jasinski, Aug 04 2025

A367312 Minimum value of 2nd derivative of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.

Original entry on oeis.org

0, 6, 7, 4, 1, 9, 2, 5, 9, 6, 9, 6, 7, 5, 6, 0, 7, 2, 5, 4, 7, 5, 3, 0, 6, 6, 6, 9, 2, 6, 7, 3, 0, 4, 6, 7, 1, 0, 1, 3, 0, 8, 6, 8, 9, 9, 9, 8, 9, 0, 1, 2, 8, 0, 8, 7, 2, 2, 2, 1, 2, 2, 4, 9, 1, 5, 0, 2, 5, 3, 5, 5, 4, 3, 6, 4, 6, 7, 3, 4, 1, 7, 4, 5, 9, 6, 2

Offset: 0

Clark Kimberling and Peter J. C. Moses, Nov 13 2023

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to (1 - 2^(1-x)) zeta(x) (0,1). This series can be described as an alternating version of the "p-series" when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x and g(x) = (1 - 2^(1-x)) zeta(x). Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined. Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

			Minimum value of f"(x), where f(x) = (1 - 2^(1-x)) zeta(x), for 0 < x < 1:
0.0641392820642571684220887165127181687393656828446464013955957700...,
which occurs for x = 0.59737100658235275929541785444598... .

Cf. A113024, A367309, A367311.

f[x_] := (1 - 2^(1 - x)) Zeta[x];
y = FindMinimum[{f''[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
RealDigits[y][[1]][[1]]

A367409 Decimal expansion of arclength of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.

Original entry on oeis.org

1, 0, 1, 8, 6, 5, 6, 3, 5, 1, 6, 7, 4, 0, 5, 1, 3, 6, 7, 3, 6, 6, 2, 2, 9, 9, 2, 5, 2, 5, 2, 7, 5, 4, 5, 3, 4, 0, 2, 6, 6, 2, 2, 5, 5, 1, 2, 4, 5, 0, 1, 7, 5, 9, 5, 0, 9, 8, 6, 2, 0, 3, 0, 5, 7, 2, 0, 6, 3, 0, 7, 5, 2, 3, 7, 7, 8, 9, 5, 9, 9, 6, 6, 9, 8, 1

Offset: 1

Clark Kimberling, Nov 26 2023

See A367309.

			1.0186563516740513673662299252527545340266...

Cf. A113024, A367309, A367311.

f[x_] := (1 - 2^(1 - x)) Zeta[x]
y = NIntegrate[Sqrt[1 + f'[x]^2], {x, 0, 1}, WorkingPrecision -> 200]
RealDigits[y][[1]]

f(x) = (1 - 2^(1-x))*zeta(x); intnum(x=0, 1, sqrt(1+f'(x)^2)) \\ Michel Marcus, Nov 27 2023

cp's OEIS Frontend

A367310 Continued fraction of the constant in A367309 (the area under the curve (1 - 2^(1-x)) zeta(x) from 0 to 1).

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A367311 Maximum curvature of the curve (1 - 2^(1-x)) zeta(x) from 0 to 1.

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A367312 Minimum value of 2nd derivative of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.

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Mathematica

A367409 Decimal expansion of arclength of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.

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PARI