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.

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, 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, 8, 0, 9, 6, 8, 4, 9, 1, 6, 0, 4, 0, 6, 0, 1, 1

Offset: 1

Author

Roberto Trocchi, Jun 21 2024

Keywords

• nonn
• cons

Comments

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.

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

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]

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

Examples

			4.9068764351428513475351082583558535315328564648993...
		

Links

• Roberto Trocchi, The Psi function and its zeros on the complex plane, June 21 2024.

Crossrefs

Cf. A091131, A058303.

Programs

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]]

Formula

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