A367312 - OEIS

A367312

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

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

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OFFSET

0,2

COMMENTS

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... .

LINKS

Table of n, a(n) for n=0..86.

EXAMPLE

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... .

MATHEMATICA

f[x_] := (1 - 2^(1 - x)) Zeta[x];

y = FindMinimum[{f''[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]

RealDigits[y][[1]][[1]]

CROSSREFS

Cf. A113024, A367309, A367311.