cp's OEIS Frontend

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

%I A367312 #14 Dec 23 2023 14:42:49
%S A367312 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 A367312 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 A367312 5,3,5,5,4,3,6,4,6,7,3,4,1,7,4,5,9,6,2
%N A367312 Minimum value of 2nd derivative of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.
%C A367312 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 A367312 Minimum value of f"(x), where f(x) = (1 - 2^(1-x)) zeta(x), for 0 < x < 1:
%e A367312 0.0641392820642571684220887165127181687393656828446464013955957700...,
%e A367312 which occurs for x = 0.59737100658235275929541785444598... .
%t A367312 f[x_] := (1 - 2^(1 - x)) Zeta[x];
%t A367312 y = FindMinimum[{f''[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
%t A367312 RealDigits[y][[1]][[1]]
%Y A367312 Cf. A113024, A367309, A367311.
%K A367312 nonn,cons
%O A367312 0,2
%A A367312 _Clark Kimberling_ and _Peter J. C. Moses_, Nov 13 2023