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

%I A379605 #10 Dec 30 2024 09:18:02
%S A379605 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,
%T A379605 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,
%U A379605 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
%N A379605 Decimal expansion of sigma_sup = sup{real(s): Psi(s) = 0}, where Psi(s) = Sum_{n>=1} 1/n!^s.
%C A379605 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}.
%C A379605 The value is approximately 0.726347508576.
%C A379605 So all the zeros of the Psi function seem to be in the critical strip 0 < real(s) < sigma_sup.
%C A379605 See my document on the zeros of the Psi function on the complex plane.
%H A379605 MathOverflow.net, <a href="https://mathoverflow.net/q/473764">The location of the zeros of the "new" function Psi</a>.
%H A379605 Roberto Trocchi, <a href="https://digilander.libero.it/tr7mail/Psi_function_zeros_v2.0.pdf">The Psi function and its zeros on the complex plane - Version 2.0</a>, December 27 2024.
%F A379605 sigma_sup = sup{real(s): Psi(s) = 0}, where Psi(s) = Sum_{n>=1} 1/n!^s.
%e A379605 0.726347508576201145941640262269523250850134334300641278184683634...
%t A379605 Nmax = 200;
%t A379605 Cn = {1}; kn = {0};
%t A379605 For[n = 2, n <= Nmax, n = n + 1,
%t A379605  If[PrimeQ[n],
%t A379605   If[Cn[[n - 1]] == 1, AppendTo[kn, 1], AppendTo[kn, 0]];
%t A379605   AppendTo[Cn, -1], PF = FactorInteger[n];
%t A379605   For[m = 1; somma = 0, m <= Length[PF], m = m + 1,
%t A379605    somma = somma + kn[[PF[[m]][[1]]]]*PF[[m]][[2]]];
%t A379605   AppendTo[kn, Mod[somma, 2]];
%t A379605   If[kn[[n]] == 0, AppendTo[Cn, Cn[[n - 1]]],
%t A379605    AppendTo[Cn, -Cn[[n - 1]]]]]]
%t A379605 NSolveValues[ {Sum[Cn[[n]]*n!^-sigma, {n, 1, Nmax}] == 0,
%t A379605    sigma > 1/10, sigma < 1}, sigma, WorkingPrecision -> 200][[1]]
%Y A379605 Cf. A373204.
%K A379605 nonn,cons
%O A379605 0,1
%A A379605 _Roberto Trocchi_, Dec 27 2024