cp's OEIS Frontend

A373204 Decimal expansion of the imaginary part of the first zero, for real(s) >= 1/2, of the function Psi(s) = Sum_{n>=1} 1/n!^s.

%I A373204 #32 Dec 30 2024 09:22:13
%S A373204 4,9,0,6,8,7,6,4,3,5,1,4,2,8,5,1,3,4,7,5,3,5,1,0,8,2,5,8,3,5,5,8,5,3,
%T A373204 5,3,1,5,3,2,8,5,6,4,6,4,8,9,9,3,3,7,6,3,5,2,0,2,8,8,9,5,2,4,8,7,0,0,
%U A373204 8,0,9,6,8,4,9,1,6,0,4,0,6,0,1,1
%N A373204 Decimal expansion of the imaginary part of the first zero, for real(s) >= 1/2, of the function Psi(s) = Sum_{n>=1} 1/n!^s.
%C A373204 Defining the Psi function to be Psi(s) = Sum_{n>=1} 1/n!^s, the first zero, for real(s) >= 1/2, is approximately s1 = 0.6418158643 + 4.9068764351*i.
%C A373204 All the zeros of the Psi function seem (conjecturally) to be in the critical strip 0 < real(s) <= 1.
%C A373204 Moreover, all the zeros of the Psi function seem (conjecturally) to be in the strip 0 < real(s) <= 0.73. [There is obviously something wrong here! - _N. J. A. Sloane_, Dec 30 2024]
%C A373204 See my document on the zeros of the Psi function on the complex plane.
%H A373204 Roberto Trocchi, <a href="https://digilander.libero.it/tr7mail/Psi_function_zeros.pdf">The Psi function and its zeros on the complex plane</a>, June 21 2024.
%F A373204 Imaginary part of the first zero for real(s) >= 1/2, Psi(s) = 0, where Psi(s) = Sum_{n>=1} 1/n!^s.
%e A373204 4.9068764351428513475351082583558535315328564648993...
%t A373204 Psi[s_, nmax_] := ParallelSum[1/n!^s, {n, 1, nmax}]
%t A373204 FindRoot[{Re[Psi[x + y*I, 2000]], Im[Psi[x + y*I, 2000]]}, {{x, 1/2}, {y, 5}}, WorkingPrecision -> 1000][[2]][[2]]
%Y A373204 Cf. A091131, A058303.
%K A373204 nonn,cons
%O A373204 1,1
%A A373204 _Roberto Trocchi_, Jun 21 2024