OFFSET
1,1
COMMENTS
Defining the Psi function as Psi(s) = Sum_{n>=1} n!^s, the first zero, for real(s) >= 1/2, is approximately s1 = 0.6418158643 + 4.9068764351*i.
All the zeros of the Psi function seem (conjecture) to be in the critical strip 0 < real(s) <= 1.
Moreover, all the zeros of the Psi function seem (conjecture) to be in the strip 0 < real(s) <= 0.73.
See my document on the zeros of the Psi function on the complex plane.
LINKS
R. Trocchi, The Psi function and its zeros on the complex plane, June 21 2024.
FORMULA
Imaginary part of the first zero for real(s) >= 1/2, Psi(s) = 0, where Psi(s) = Sum_{n>=1} n!^s.
EXAMPLE
4.9068764351428513475351082583558535315328564648993...
MATHEMATICA
Psi[s_, nmax_] := ParallelSum[1/n!^s, {n, 1, nmax}]
FindRoot[{Re[Psi[x + y*I, 2000]], Im[Psi[x + y*I, 2000]]}, {{x, 1/2}, {y, 5}}, WorkingPrecision -> 1000][[2]][[2]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Roberto Trocchi, Jun 21 2024
STATUS
approved