A074760 -id:A074760 - cp's OEIS frontend

A332645 Decimal expansion of Sum_{n>=1} 1/z(n)^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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

Offset: 0

Artur Jasinski, Feb 18 2020

			0.0231049931154189707889338104303390140033817603974220901231825...

J. P. Gram, "Note sur le calcul de la fonction zeta(s) de Riemann", Det Kgl. Danske Vid. Selsk. Overs., 1895, pp. 303-308. p.307 (16 decimal digits).
Charles Jean De La Vallée Poussin, Sur La Fonction de Riemann Et Le Nombre Des Nombres Premiers Inférieurs à Une Limite Donnee, 1899.

Jesús Guillera, Some sums over the non-trivial zeros of the Riemann zeta function, arXiv:1307.5723v7 [math.NT], 2013-2014; see p.9 eq.(21).
Kano Kono, Vieta's Formulas on Completed Riemann Zeta, Alien's Mathematics p. 13 eq. 2.3(2).
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen I, 1909, pp. 310-342.
J. J. Y. Liang and John Todd, The Stieltjes Constants, Journal of Research of the National Bureau of Standards, 1972, pp. 175-176.
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. A006752, A074760, A332614.

evalf((-32 - log(Pi)^2 + Psi(0, 1/4)^2 + Psi(1, 1/4) + 4*(Psi(0, 1/4) * Zeta(1, 1/2) + Zeta(2, 1/2)) / Zeta(1/2)) / 8, 120); # Vaclav Kotesovec, Feb 19 2020

Join[{0}, RealDigits[N[-4 + Catalan + Pi^2/8 + (Zeta''[1/2]/Zeta[1/2] - (Zeta'[1/2] / Zeta[1/2])^2)/2, 105]][[1]]]
N[SeriesCoefficient[Log[s*(s-1)*Pi^(-s/2)*Gamma[s/2]*Zeta[s]/2], {s, 1/2, 2}], 105] (* Vaclav Kotesovec, Feb 19 2020 *)

Equals -4 + G + Pi^2/8 + (1/2)(zeta''(1/2)/zeta(1/2) - (zeta'(1/2)/zeta(1/2))^2) where G is the Catalan constant A006752.
Equals G - 4 + (Pi^2 - (gamma + Pi/2 + log(8*Pi))^2) / 8 + zeta''(1/2) / (2*zeta(1/2)), where gamma is the Euler-Mascheroni constant A001620 and G is the Catalan constant A006752. - Vaclav Kotesovec, Feb 19 2020
Also equals (-32 - log(Pi)^2 + psi(0, 1/4)^2 + psi(1, 1/4) + 4*(psi(0, 1/4) * zeta'(1/2) + zeta''(1/2)) / zeta(1/2)) / 8, where psi(0, 1/4) = -A020777 and psi(1, 1/4) = A282823. - Vaclav Kotesovec, Feb 19 2020

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.

Original entry on oeis.org

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).

A332614 a(n) is the smallest index k such that Sum_{m=1..k} 1/z(m) > n where z(m) is the imaginary part of the m-th nontrivial zero of the Riemann zeta function, n=0,1,2,...

Original entry on oeis.org

1, 93, 621, 2437, 7438, 19490, 45996, 100462, 206617, 404855, 762155, 1387088, 2452209, 4227039, 7126088, 11778044, 19124514, 30559702, 48126380, 74788784, 114809974, 174270215, 261774713, 389414312, 574062463, 839117171, 1216829213, 1751399577, 2503082172, 3553595368

Offset: 0

Artur Jasinski, Feb 17 2020

nonn

Because series Sum_{m>=1} 1/z(m) is divergent this sequence is infinite.
Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.
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.
a(11)-a(39) computed by David Platt, Mar 20 2020.

			a(0)=1 because 1/z(1) = 0.070747749954285585596 > 0
a(1)=93 because Sum_{m=1..93} 1/z(m) = 1.00082895080028509266 > 1
a(2)=621 because Sum_{m=1..621} 1/z(m) = 2.00017203211984838994 > 2.

Artur Jasinski, Table of n, a(n) for n = 0..39
Kano Kono, Vieta's Formulas on Completed Riemann Zeta, Alien's Mathematics.

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

aa = {}; kk = 0; b = 0; Do[b = b + N[1/Im[ZetaZero[n]], 30];
If[b > kk, AppendTo[aa, n]; kk = kk + 1];, {n, 1, 1000000}]; aa

More terms from Artur Jasinski, Feb 21 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).

A195423 Decimal expansion of -2B = sum(r in Z, 1/(r*(1-r))), where Z is the set of zeros of the Riemann zeta function which lie in the strip 0 <= Re(z) <= 1.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Sep 18 2011

See A074760 for references and links.

			-2B = gamma + 2 - log(4*Pi) = 0.046191417932242...

Cf. A002410, A074760, A233825.

RealDigits[ N[ EulerGamma + 2 - Log[4*Pi], 105], 10, 100] [[1]]

Euler-log(4*Pi)+2 \\ Charles R Greathouse IV, Mar 10 2016

As a constant, equals 2*A074760.

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.

A246843 Decimal expansion of C, a constant associated with the estimation of the maximum of |zeta(1+i*t)|.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 05 2014

			-0.089326522343551...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 28.
Tadej Kotnik, Computational estimation of the constant β(1) characterizing the order of ζ(1 + it), Math. Comp. 77: 1713-1723, 2008.

Cf. A074760, A104539, A104540, A242056.

digits = 15; precision = 200; u0 = 10^8; du = 10^8; tail[u_] := -(1 + Log[2*Pi*u])/(2*u); Clear[f]; f[u_] := f[u] = 1 - Log[2] + NIntegrate[Log[BesselI[0, t]]/t^2, {t, 0, 2} , WorkingPrecision -> precision] + NIntegrate[(Log[BesselI[0, t]] - t)/t^2, {t, 2, u}, WorkingPrecision -> precision, MaxRecursion -> 20 ] + tail[u]; f[u0]; f[u = u0 + du]; While[RealDigits[f[u], 10, digits + 4] != RealDigits[f[u - du], 10, digits + 4], Print["u = ", u, " ", f[u]]; u = u + du]; Join[{0}, RealDigits[f[u], 10, digits] // First]

1 - log(2) + integral_{0..2} log(BesselI(0, t))/t^2 dt + integral_{2..infinity} (log(BesselI(0, t)) - t)/t^2 dt.

Typo in the formula corrected by Vaclav Kotesovec, Sep 17 2014

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.

cp's OEIS Frontend

A332645 Decimal expansion of Sum_{n>=1} 1/z(n)^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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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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A332614 a(n) is the smallest index k such that Sum_{m=1..k} 1/z(m) > n where z(m) is the imaginary part of the m-th nontrivial zero of the Riemann zeta function, n=0,1,2,...

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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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A195423 Decimal expansion of -2B = sum(r in Z, 1/(r*(1-r))), where Z is the set of zeros of the Riemann zeta function which lie in the strip 0 <= Re(z) <= 1.

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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.