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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.
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%I #32 Dec 30 2024 09:22:13

%S 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 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 8,0,9,6,8,4,9,1,6,0,4,0,6,0,1,1

%N 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 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 All the zeros of the Psi function seem (conjecturally) to be in the critical strip 0 < real(s) <= 1.

%C 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 See my document on the zeros of the Psi function on the complex plane.

%H 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 Imaginary part of the first zero for real(s) >= 1/2, Psi(s) = 0, where Psi(s) = Sum_{n>=1} 1/n!^s.

%e 4.9068764351428513475351082583558535315328564648993...

%t Psi[s_, nmax_] := ParallelSum[1/n!^s, {n, 1, nmax}]

%t FindRoot[{Re[Psi[x + y*I, 2000]], Im[Psi[x + y*I, 2000]]}, {{x, 1/2}, {y, 5}}, WorkingPrecision -> 1000][[2]][[2]]

%Y Cf. A091131, A058303.

%K nonn,cons

%O 1,1

%A _Roberto Trocchi_, Jun 21 2024