Roberto Trocchi - cp's OEIS frontend

User: Roberto Trocchi

Roberto Trocchi has authored 3 sequences.

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

Roberto Trocchi, Dec 27 2024

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.

			0.726347508576201145941640262269523250850134334300641278184683634...

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.

Cf. A373204.

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]]

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

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.

Original entry on oeis.org

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

Roberto Trocchi, Jun 21 2024

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.

			4.9068764351428513475351082583558535315328564648993...

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

Cf. A091131, A058303.

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]]

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

A370697 Decimal expansion of Sum_{p prime} 1/p!^p.

Original entry on oeis.org

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

Offset: 0

Roberto Trocchi, Jun 07 2024

			0.25462962966981738683127573227020697042409997891314217843798476619...

Cf. A068210, A100124, A094289, A073009.

RealDigits[Sum[1/Prime[n]!^Prime[n], {n, 1, 12}], 10, 1000] // First

suminf(k=1,1/prime(k)!^prime(k)) \\ Hugo Pfoertner, Jun 22 2024

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User: Roberto Trocchi

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

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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.

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A370697 Decimal expansion of Sum_{p prime} 1/p!^p.

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