A333360 -id:A333360 - cp's OEIS frontend

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.

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).

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.

A360807 Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2) where z(m) is the imaginary part of the m-th nontrivial zero of the Dirichlet beta function whose real part is 1/2.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Feb 21 2023

Conjecture: Nontrivial zeros whose real part is not 1/2 do not exist.

			0.077783989961792964431079...

Kano Kono, Summary Dirichlet beta function, Alien's Mathematics, p. 5.

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

kk = RealDigits[N[4 Log[Gamma[3/4]] + EulerGamma/2 + Log[2] - 3 Log[Pi]/2, 115]][[1]]; Prepend[kk, 0]

4*log(gamma(3/4)) + Euler/2 + log(2) - 3*log(Pi)/2 \\ Michel Marcus, Mar 15 2023

Equals 4*log(Gamma(3/4)) + A001620/2 + log(2) - 3*log(Pi)/2.
Equals A074760 - 1 + log(4) - log(Pi) + 4*log(Gamma(3/4)).
Equals 1 - A074760 + A001620 - 2*log(Pi) + 4*log(Gamma(3/4)).
Equals 3*A074760 - 3 - A001620 + 4*log(2) + 4*log(Gamma(3/4)).

cp's OEIS Frontend

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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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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