A367312 - OEIS

%I #14 Dec 23 2023 14:42:49

%S 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,

%T 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,

%U 5,3,5,5,4,3,6,4,6,7,3,4,1,7,4,5,9,6,2

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

%C 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... .

%e Minimum value of f"(x), where f(x) = (1 - 2^(1-x)) zeta(x), for 0 < x < 1:

%e 0.0641392820642571684220887165127181687393656828446464013955957700...,

%e which occurs for x = 0.59737100658235275929541785444598... .

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

%t y = FindMinimum[{f''[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]

%t RealDigits[y][[1]][[1]]

%Y Cf. A113024, A367309, A367311.

%K nonn,cons

%O 0,2

%A _Clark Kimberling_ and _Peter J. C. Moses_, Nov 13 2023