cp's OEIS Frontend

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

%I A367311 #24 Aug 06 2025 09:05:39
%S A367311 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,
%T A367311 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,
%U A367311 5,7,6,2,7,9,8,3,6,9,3,2,1,7,2,4,9,4,7
%N A367311 Maximum curvature of the curve (1 - 2^(1-x)) zeta(x) from 0 to 1.
%C A367311 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 A367311 Maximum curvature = 0.0641392820642571684220887165127181687393..., which occurs at x = 0.6827548440370203586269... .
%t A367311 f[x_] := (1 - 2^(1 - x)) Zeta[x];
%t A367311 c[x_] := Abs[f''[x]]/(1 + f'[x]^2)^(3/2)
%t A367311 y = FindMaximum[{c[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
%t A367311 RealDigits[y][[1]][[1]]
%Y A367311 Cf. A113024, A367309, A367312.
%K A367311 nonn,cons
%O A367311 0,2
%A A367311 _Clark Kimberling_ and _Peter J. C. Moses_, Nov 13 2023
%E A367311 One initial 0 inserted by _Artur Jasinski_, Aug 04 2025