A335814 -id:A335814 - 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).

A335815 Decimal expansion of Sum_{n>=1} 1/z(n)^4 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, 3, 7, 1, 7, 2, 5, 9, 9, 2, 8, 5, 2, 6, 9, 6, 8, 6, 1, 6, 4, 8, 6, 6, 2, 6, 2, 4, 7, 1, 7, 4, 0, 5, 7, 8, 4, 5, 3, 6, 5, 0, 8, 8, 9, 7, 3, 0, 0, 8, 3, 2, 1, 3, 5, 7, 5, 5, 0, 6, 3, 7, 1, 8, 4, 6, 1, 3, 3, 2, 9, 8, 8, 4, 5, 7, 2, 8, 1, 3, 7, 2, 9, 7, 6, 0, 3, 5, 7, 2, 3, 3, 7, 4, 2, 4, 2, 9, 6, 0, 2, 8, 3, 7, 0, 0

Offset: 0

Artur Jasinski, Jun 25 2020

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 = 0.0007295482727097...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; this sequence.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
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.

			0.0000371725992852696861648662624717405784536508897300...

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, p. 665-699.
André Voros, Zeta functions over Zeros of the Zeta functions, 2010, p. 153.

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

Join[{0,0,0,0},RealDigits[N[-1/12*(D[Log[Zeta[x]],{x,4}]/. x -> 1/2) - 1/24 Pi^4 -(Zeta[4, 1/4] - Zeta[4, 3/4])/64 + 16, 105]][[1]]]

Equals 16-Pi^4/24+(Zeta[4,3/4]-Zeta[4,1/4])/64-(Log[Zeta[x]]''''[1/2])/24

A335826 Decimal expansion of Sum_{n>=1} 1/z(n)^6 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, 0, 1, 4, 4, 1, 7, 3, 9, 3, 1, 4, 0, 0, 9, 7, 3, 2, 7, 9, 6, 9, 5, 3, 8, 1, 5, 5, 6, 0, 9, 4, 8, 2, 0, 9, 0, 7, 0, 3, 6, 8, 8, 3, 0, 0, 8, 5, 0, 9, 0, 9, 8, 1, 1, 8, 7, 1, 5, 9, 9, 9, 3, 6, 4, 2, 1, 7, 9, 0, 5, 3, 9, 4, 6, 3, 1, 6, 8, 9, 6, 4, 0, 8, 1, 9, 5, 5, 0, 6, 7, 4, 2, 0, 4, 6, 8, 3, 8, 8, 8, 3, 4, 2, 3, 0, 5

Offset: 0

Artur Jasinski, Jun 25 2020

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 = 0.0007295482727097...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
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.

			0.000000144173931400973279695381556....

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, p. 665-699.
André Voros, Zeta functions over Zeros of the Zeta functions, 2010, p. 153.

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

m = 3; Join[{0, 0, 0, 0, 0, 0},RealDigits[N[((-1)^m (2^(2 m) - ((2^(2 m) - 1) Zeta[2 m] + (Zeta[2 m, 1/4] - Zeta[2 m, 3/4])/2^(2 m))/4 - (D[Log[Zeta[x]], {x, 2 m}] /. x -> 1/2)/(2 (2 m - 1)!) )), 105]][[1]]]

Universal formula for Sum_{n>=1} 1/z(n)^(2m) published in Voros 2002-2003 p. 22 (see Mathematica procedure below).

A355283 Decimal expansion of the constant B(3) = Sum_{n>=1} Sum_{m>=n+1} 1/(z(n)*z(m))^3 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, 0, 1, 9, 4, 0, 3, 3, 3, 7, 5, 4, 0, 6, 3, 6, 9, 8, 3, 6, 7, 2, 7, 2, 4, 8, 7, 9, 8, 1, 5, 4, 7, 5, 0, 6, 6, 4, 5, 0, 0, 6, 4, 5, 6, 7, 0, 1, 0, 0, 0, 0, 1, 0, 8, 9, 6, 8, 8, 8, 7, 7, 9, 5, 3, 1, 0, 3, 1, 0, 9, 3, 5, 3, 2, 5, 7, 7, 2, 6, 0, 6, 5, 8, 0, 3, 8, 6, 3, 6, 8, 8, 3, 1, 7, 5, 3, 5, 1, 5, 1, 8, 8, 4, 4, 6, 0, 5, 1, 7, 4

Offset: 0

Artur Jasinski, Aug 20 2022

			0.0000001940333754063698367... = 1.940333754063698367*10^(-7).

For links see A332645.

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

Equals (A333360^2 - A335826)/2.

No simpler formula is known.

A337365 Decimal expansion of imaginary part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 26 2020

For the decimal expansion of the real part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function see A337404.
Sum_{m>=1} 1/(1/2 + i*z(m))^1 = 0.01154785448306... - i*A where 0.01154785448306 = A074760/2 and A > 10.5.
Sum_{m>=1} 1/(1/2 + i*z(m))^2 = -0.0230771586479... - i*0.000728434... where -0.0230771586479 = A245275/2
Sum_{m>=1} 1/(1/2 + i*z(m))^3 = -0.000055579115726... + i*0.0007262105... where -0.000055579115726 = A245276/2
Sum_{m>=1} 1/(1/2 + i*z(m))^4 = 0.0000368136106308... + i*0.0000044382...
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 = 0.0007295482727097...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
Sum_{m>=1} 1/z(m)^6 = 0.0000001441739314...; 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

			0.000004438269312506953

See A332645.

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

(* 7-day-long procedure *)
kk = 0; Do[kk = kk + 1/(N[ZetaZero[n], 100])^4 , {n, 1, 1000000}]; Take[Join[{0, 0, 0, 0, 0}, RealDigits[Im[kk]][[1]]], 11]

No explicit formula is known.

A337404 Decimal expansion of real part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 26 2020

For the decimal expansion of the imaginary part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function see A337365.

A335918 Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2)^2 where z(m) is the imaginary part of the m-th nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jun 29 2020

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 = 0.0007295482727097...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
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.

			0.0000371006364374648715125054339132797135962919799565287...

André Voros, Zeta functions for the Riemann zeros, arXiv:math/0104051 [math.CV], 2002-2003, p.22 Table 1.

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

Join[{0, 0, 0, 0},RealDigits[N[3 + EulerGamma + EulerGamma^2 - Pi^2/8 - Log[4 Pi] + 2 StieltjesGamma[1], 105]][[1]]]

Equals: 3 + gamma + gamma^2 - Pi^2/8 - log(4*Pi) + 2*gamma(1), where gamma is the Euler-Mascheroni gamma constant (see A001620) and gamma(1) is 1st Stieltjes constant (see A082633).

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.

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.

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A335815 Decimal expansion of Sum_{n>=1} 1/z(n)^4 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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A335826 Decimal expansion of Sum_{n>=1} 1/z(n)^6 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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A355283 Decimal expansion of the constant B(3) = Sum_{n>=1} Sum_{m>=n+1} 1/(z(n)*z(m))^3 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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A337365 Decimal expansion of imaginary part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).

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A337404 Decimal expansion of real part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).

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A335918 Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2)^2 where z(m) is the imaginary part of the m-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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