A355283 -id:A355283 - cp's OEIS frontend

A333360 Decimal expansion of Sum_{n>=1} 1/z(n)^3 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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

Offset: 0

Artur Jasinski, Mar 16 2020

a(1)-a(7) published by André Voros in 2001.
a(8)-a(20) computed by David Platt, Mar 15 2020.
a(21)-a(78) computed by Fredrik Johansson, Aug 04 2022 by mpmath procedure.
a(79)-a(350) computed by Artur Kawalec, Aug 15 2022 up to 350 decimal digits on basis algorithm of Juan Arias de Reyna.
a(351)-a(495) computed by Juan Arias de Reyna, using his implementation in mpmath from 2010, documented in his paper from 2020 (see link).
b-file on basis data from email Aug 16 2022 of Juan Arias de Reyna to Artur Jasinski.
Sum_{m>=1} 1/z(m) is a divergent series; see A332614.
Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.
Sum_{m>=1} 1/z(m)^3 = 0.0007295482727...; this sequence.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992...; see A335815.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886...; see A335814.
Sum_{m>=1} 1/z(m)^6 = 0.0000001441739...; see A335826.
Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.
Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.
Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276.
Sum_{r>=1} Sum_{m>=n+1} 1/(z(r)*z(m))^3 = 0.00000619403... see A355283.

			0.00072954827270970421...

Artur Jasinski, Table of n, a(n) for n = 0..497
mpmath, Online versions.
Artur Kawalec, The recurrence formulas for primes and non-trivial zeros of the Riemann zeta function, arxiv:2009.02640 [math.NT], 2020.
Artur Kawalec, Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function, arxiv:2012.06581 [math.NT], 2021.
Artur Kawalec, The inverse Riemann zeta function, arxiv:2106.06915 [math.NT], 2021, p. 38 formula (146).
Juan Arias de Reyna, Computation of the secondary zeta function, arxiv:2006.04869 [math.NT], 2020.
André Voros, Zeta functions for the Riemann zeros, arXiv:math/0104051 [math.CV], 2002-2003, p. 25 Table 2.
André Voros, Zeta functions for the Riemann zeros, 2001(2008) p. 20 Table 1.
André Voros, Zeta functions for the Riemann zeros, Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 665-699.

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645.

from mpmath import *
mp.dps = 90
nprint(secondzeta(3), 78)

No explicit formula is known (André Voros, personal communication to Artur Jasinski, Mar 09 2020).

A335814 Decimal expansion of Sum_{n>=1} 1/z(n)^5 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jun 25 2020

a(1)-a(34) computed by David Platt, Mar 15 2020.
a(35)-a(78) computed by Fredrik Johansson, Aug 04 2022 by mpmath procedure.
a(79)-a(115) computed by Artur Kawalec, Aug 15 2022 up to 350 decimal digits on basis Juan Arias de Reyna algorithm.
b-file on basis data from email Aug 15 2022 from Artur Kawalec to Artur Jasinski.
Sum_{m>=1} 1/z(m) is a divergent series; see A332614.
Sum_{m>=1} 1/z(m)^2 = 0.0231049931154...; see A332645.
Sum_{m>=1} 1/z(m)^3 = 0.0007295482727...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992...; see A335815.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886...; see A335814.
Sum_{m>=1} 1/z(m)^6 = 0.0000001441739...; see A335826.
Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.
Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.
Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276.
Sum_{r>=1} Sum_{m>=n+1} 1/(z(r)*z(m))^3 = 0.00000619403... see A355283.

			0.0000022311886995021033286406286918...

Artur Jasinski, Table of n, a(n) for n = 0..350
Artur Kawalec, The recurrence formulas for primes and non-trivial zeros of the Riemann zeta function, arxiv:2009.02640 [math.NT], 2020.
Artur Kawalec, Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function, arxiv:2012.06581 [math.NT], 2021.
Artur Kawalec, The inverse Riemann zeta function, arxiv:2106.06915 [math.NT], 2021 p. 38 formula (146).
Juan Arias de Reyna, Computation of the secondary zeta function, arxiv:2006.04869 [math.NT], 2020.

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815, A335826, A355283.

from mpmath import *
mp.dps = 90
nprint(secondzeta(5), 78)

No explicit formula for Sum_{n>=1} 1/z(n)^k is known for odd exponents k (André Voros, personal communication to Artur Jasinski, Mar 09 2020).

A356693 Decimal expansion of the constant B(2) = Sum_{n>=1} Sum_{m>=n+1} 1/(z(n)*z(m))^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 23 2022

			0.000248334053789144...

Cf. A355283 (B(3)).

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815, A355283.

Join[{0, 0, 0}, RealDigits[N[-4*Catalan + Catalan^2/2 - Pi^2/2 + (Catalan*Pi^2)/8 + Pi^4/128 + (1/64)*Zeta[4, 1/4] + (2*Zeta'[1/2]^2)/Zeta[1/2]^2 - (Catalan Zeta'[1/2]^2)/(2 Zeta[1/2]^2) - (Pi^2 Zeta'[1/2]^2)/(16*Zeta[1/2]^2) - Zeta'[1/2]^4/(8*Zeta[1/2]^4) - (2 Zeta''[1/2])/Zeta[1/2] + (Catalan Zeta''[1/2])/(2 Zeta[1/2]) + (Pi^2 Zeta''[1/2])/(16*Zeta[1/2]) + Zeta'[1/2]^2*Zeta''[1/2]/(4 Zeta[1/2]^3) - Zeta'[1/2] Zeta'''[1/2]/(6 Zeta[1/2]^2) + Zeta''''[1/2]/(24  Zeta[1/2]), 115]][[1]]]

Equals (A332645^2 - A335815)/2.

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A333360 Decimal expansion of Sum_{n>=1} 1/z(n)^3 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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A335814 Decimal expansion of Sum_{n>=1} 1/z(n)^5 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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A356693 Decimal expansion of the constant B(2) = Sum_{n>=1} Sum_{m>=n+1} 1/(z(n)*z(m))^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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