A379605 Decimal expansion of sigma_sup = sup{real(s): Psi(s) = 0}, where Psi(s) = Sum_{n>=1} 1/n!^s.

7, 2, 6, 3, 4, 7, 5, 0, 8, 5, 7, 6, 2, 0, 1, 1, 4, 5, 9, 4, 1, 6, 4, 0, 2, 6, 2, 2, 6, 9, 5, 2, 3, 2, 5, 0, 8, 5, 0, 1, 3, 4, 3, 3, 4, 3, 0, 0, 6, 4, 1, 2, 7, 8, 1, 8, 4, 6, 8, 3, 6, 3, 4, 1, 2, 6, 5, 6, 2, 9, 9, 1, 7, 8, 3, 2, 3, 2, 9, 9, 1, 1, 9, 3, 4, 0, 8, 9, 2, 3, 5, 9, 0, 6, 4, 4, 6, 9, 8, 3

Offset

0,1

Comments

Defining the Psi function to be Psi(s) = Sum_{n>=1} 1/n!^s, in the MathOverflow link I have posted the description of an algorithm to calculate the exact value of sigma_sup = sup{real(s): Psi(s) = 0}.

The value is approximately 0.726347508576.

So all the zeros of the Psi function seem to be in the critical strip 0 < real(s) < sigma_sup.

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

Links

Table of n, a(n) for n=0..99.

MathOverflow.net, The location of the zeros of the "new" function Psi.

Roberto Trocchi, The Psi function and its zeros on the complex plane - Version 2.0, December 27 2024.

Formula

sigma_sup = sup{real(s): Psi(s) = 0}, where Psi(s) = Sum_{n>=1} 1/n!^s.

Example

0.726347508576201145941640262269523250850134334300641278184683634...

Mathematica

Nmax = 200;
Cn = {1}; kn = {0};
For[n = 2, n <= Nmax, n = n + 1,
 If[PrimeQ[n],
  If[Cn[[n - 1]] == 1, AppendTo[kn, 1], AppendTo[kn, 0]];
  AppendTo[Cn, -1], PF = FactorInteger[n];
  For[m = 1; somma = 0, m <= Length[PF], m = m + 1,
   somma = somma + kn[[PF[[m]][[1]]]]*PF[[m]][[2]]];
  AppendTo[kn, Mod[somma, 2]];
  If[kn[[n]] == 0, AppendTo[Cn, Cn[[n - 1]]],
   AppendTo[Cn, -Cn[[n - 1]]]]]]
NSolveValues[ {Sum[Cn[[n]]*n!^-sigma, {n, 1, Nmax}] == 0,
   sigma > 1/10, sigma < 1}, sigma, WorkingPrecision -> 200][[1]]

Crossrefs

Cf. A373204.

Sequence in context: A387297 A309647 A225444 * A175408 A019934 A295874

Adjacent sequences: A379602 A379603 A379604 * A379606 A379607 A379608

Keyword

nonn,cons

Author

Roberto Trocchi, Dec 27 2024

Status

approved