A104540 -id:A104540 - cp's OEIS frontend

A074760 Decimal expansion of lambda(1) in Li's criterion.

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

Offset: 0

Benoit Cloitre, Sep 28 2002

Decimal expansion of -B =(1/2)*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.

According to Gun, Murty, & Rath (2018), it is not even known whether this constant is rational or not (though see Theorem 3.1), though they show that it is transcendental under Schanuel’s conjecture. - Charles R Greathouse IV, Nov 12 2021

			0.023095708966121033814310247906495291621932127152050759525392...

H. M. Edwards, Riemann's Zeta Function, Dover Publications Inc. 1974, p. 160.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.6.2, 2.21, and 2.32, pp. 42, 168, 204.
S. J. Patterson, "An introduction to the theory of the Riemann Zeta-function", Cambridge Studies in Advanced Mathematics 14, p. 34.

E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287.
M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534.
Sanoli Gun, M. Ram Murty, and Purusottam Rath, Transcendental sums related to the zeros of zeta functions, arXiv:1807.11201 [math.NT], 2018; Mathematika, Vol. 64, no. 3 (2018), pp. 875-897.
Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333.
Stephane Louboutin, Majorations explicites de |L(1, χ)| (Suite), C. R. Acad. Sci. Paris. 323, pp. 443-446 (1996). (In French). See Theorem 1 at p. 444.
J. Sondow and C. Dumitrescu, A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis, arXiv:1005.1104 [math.NT], 2010; see p. 3 in the link.
J. Sondow and C. Dumitrescu, A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis, Periodica Math. Hungarica, 60 (2010), 37-40; see p. 39 in the link.
Eric Weisstein's World of Mathematics, Li's Criterion.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Wikipedia, Li's criterion.
Index entries for zeta function.

Cf. A002410 (nearest integer to imaginary part of n-th zeta zero), A195423 (twice the constant).
Cf. A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4), A104542 (lambda_5).
Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

RealDigits[EulerGamma/2 + 1 - Log[4 Pi]/2, 10, 110][[1]]

Euler/2+1-log(4*Pi)/2 \\ Charles R Greathouse IV, Jan 26 2012

-B = Gamma/2 + 1 - log(4*Pi)/2 = 0.0230957...

Name simplified by Eric W. Weisstein, Feb 08 2019

A104539 Decimal expansion of lambda(2) in Li's criterion.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Mar 13 2005

			0.0923457352...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.32, p. 204.

E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287.
M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534.
Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333.
Eric Weisstein's World of Mathematics, Li's Criterion.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Wikipedia, Li's criterion.

Cf. A074760 (lambda_1), A104540 (lambda_3), A104541 (lambda_4), A104542 (lambda_5).

Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

lambda[n_] := Limit[D[s^(n - 1)*Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; Join[{0}, RealDigits[lambda[2], 10, 102] // First]
lambda[2] = 1 + EulerGamma - EulerGamma^2 + Pi^2/8 - Log[4 Pi] - 2*StieltjesGamma[1]; Join[{0}, RealDigits[lambda[2], 10, 102] // First] (* Jean-François Alcover, Oct 31 2012, after Eric W. Weisstein, updated May 18 2016 *)
RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + e - e^2 + Pi^2/8 - 2 g[1] - Log[4 Pi]], 10, 110, -1][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

A104541 Decimal expansion of lambda(4) in Li's criterion.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Mar 13 2005

			0.368790479...

E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287.
M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534.
Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333.
Eric Weisstein's World of Mathematics, Li's Criterion.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Wikipedia, Li's criterion.

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104542 (lambda_5).

Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

lambda[n_] := Limit[D[s^(n - 1) Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[4], 110]][[1]][[1 ;; 102]] (* Jean-François Alcover, Oct 31 2012, after Eric W. Weisstein, updated May 18 2016 *)
RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 2 e - 6 e^2 + 4 e^3 - e^4 + 3 Pi^2/4 + Pi^4/96 - 12 g[1] + 12 e g[1] - 4 e^2 g[1] - 2 g[1]^2 + 6 g[2] - 2 e g[2] - 2 g[3]/3 - 2 Log[4 Pi] - 7 Zeta[3]/2], 10, 110][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

3*Pi^2/4 + Pi^4/96 - 2*log(4) - 2*log(Pi) + 2*gamma - 6*gamma^2 + 4*gamma^3 - gamma^4 - 12*gamma(1) + 12*gamma*gamma(1) - 4*gamma^2*gamma(1) - 2*gamma(1)^2 + 6*gamma(2) - 2*gamma*gamma(2) - 2*gamma(3)/3 - 7*zeta(3)/2 + 1. - Jean-François Alcover, Jul 02 2014

A104542 Decimal expansion of lambda(5) in Li's criterion.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Mar 13 2005

			0.575542714...

E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287.
M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534.
Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333.
Eric Weisstein's World of Mathematics, Li's Criterion.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Wikipedia, Li's criterion.

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4).

Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

lambda[n_] := Limit[D[s^(n - 1)*Log[xi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[5], 110]][[1]][[1 ;; 102]] (* Jean-François Alcover, Oct 31 2012, after Eric W. Weisstein, updated May 18 2016 *)
RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 5 e/2 - 10 e^2 + 10 e^3 - 5 e^4 + e^5 + 5 Pi^2/4 + (5 Pi^4)/96 - 20 g[1] + 30 e g[1] - 20 e^2 g[1] + 5 e^3 g[1] - 10 g[1]^2 + 5 e g[1]^2 + 15 g[2] - 10 e g[2] + 5/2 e^2 g[2] + 5/2 g[1] g[2] - 10 g[3]/3 + 5/6 e g[3] + 5 g[4]/24 - Log[32] - 5 Log[Pi]/2 - 35 Zeta[3]/4 - 31 Zeta[5]/32], 10, 110][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

lambda(5) = 5*Pi^2/4 + 5*Pi^4/96 - 5*log(4)/2 - 5*log(Pi)/2 + 5*gamma/2 - 10*gamma^2 + 10*gamma^3 - 5*gamma^4+gamma^5 - 20*gamma(1) + 30*gamma*gamma(1) - 20*gamma^2*gamma(1) + 5*gamma^3*gamma(1) - 10*gamma(1)^2 + 5*gamma*gamma(1)^2 + 15*gamma(2) - 10*gamma*gamma(2) + 5/2*gamma^2*gamma(2) + 5/2*gamma(1)*gamma(2) - 10*gamma(3)/3 + 5/6*gamma*gamma(3) + 5*gamma(4)/24 - 35*zeta(3)/4 - 31*zeta(5)/32+1. - Jean-François Alcover, Jul 02 2014

A306339 Decimal expansion of lambda(6) in Li's criterion.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Feb 08 2019

			0.8275660122823792974...

E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287.
M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534.
Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333.
Eric Weisstein's World of Mathematics, Li's Criterion.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Wikipedia, Li's criterion.

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4).

Cf. A104542 (lambda_5), A306340 (lambda_7), A306341 (lambda_8).

RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 3 e - 15 e^2 + 20 e^3 - 15 e^4 + 6 e^5 - e^6 + 15 Pi^2/8 + 5 Pi^4/32 + Pi^6/960 - 30 g[1] + 60 e g[1] - 60 e^2 g[1] + 30 e^3 g[1] - 6 e^4 g[1] - 30 g[1]^2 + 30 e g[1]^2 - 9 e^2 g[1]^2 - 2 g[1]^3 + 30 g[2] - 30 e g[2] + 15 e^2 g[2] - 3 e^3 g[2] + 15 g[1] g[2] - 6 e g[1] g[2] - 3 g[2]^2/4 - 10 g[3] + 5 e g[3] - e^2 g[3] - g[1] g[3] + 5 g[4]/4 - e g[4]/4 - g[5]/20 - 3 Log[4 Pi] - 35 Zeta[3]/2 - 93 Zeta[5]/16], 10, 110][[1]]

A306340 Decimal expansion of lambda(7) in Li's criterion.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Feb 08 2019

			1.124460117570959490...

E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287.
M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534.
Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333.
Eric Weisstein's World of Mathematics, Li's Criterion.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Wikipedia, Li's criterion.

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3).

Cf. A104541 (lambda_4), A104542 (lambda_5), A306339 (lambda_6), A306341 (lambda_8).

RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 7 e/2 - 21 e^2 + 35 e^3 - 35 e^4 + 21 e^5 - 7 e^6 + e^7 + 21 Pi^2/8 + 35 Pi^4/96 + 7 Pi^6/960 - 42 g[1] + 105 e g[1] - 140 e^2 g[1] + 105 e^3 g[1] - 42 e^4 g[1] + 7 e^5 g[1] - 70 g[1]^2 + 105 e g[1]^2 - 63 e^2 g[1]^2 + 14 e^3 g[1]^2 - 14 g[1]^3 + 7 e g[1]^3 + 105 g[2]/2 - 70 e g[2] + 105/2 e^2 g[2] - 21 e^3 g[2] + 7/2 e^4 g[2] + 105/2 g[1] g[2] - 42 e g[1] g[2] + 21/2 e^2 g[1] g[2] + 7/2 g[1]^2 g[2] - 21 g[2]^2/4 + 7/4 e g[2]^2 - 70 g[3]/3 + 35/2 e g[3] - 7 e^2 g[3] + 7/6 e^3 g[3] - 7 g[1] g[3] + 7/3 e g[1] g[3] + 7/12 g[2] g[3] + 35 g[4]/8 - 7/4 e g[4] + 7/24 e^2 g[4] + 7/24 g[1] g[4] - 7 g[5]/20 + 7/120 e g[5] + 7 g[6]/720 - 7/2 Log[4 Pi] - 245 Zeta[3]/8 - 651 Zeta[5]/32 - 127 Zeta[7]/128], 10, 110][[1]]

A306341 Decimal expansion of lambda(8) in Li's criterion.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Feb 08 2019

			1.465755677147060632...

E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287.
M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534.
Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333.
Eric Weisstein's World of Mathematics, Li's Criterion.
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.
Wikipedia, Li's criterion.

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4).

Cf. A104542 (lambda_5), A306339 (lambda_6), A306340 (lambda_7).

RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 4 e - 28 e^2 + 56 e^3 - 70 e^4 + 56 e^5 - 28 e^6 + 8 e^7 - e^8 + 7 Pi^2/2 + 35 Pi^4/48 + 7 Pi^6/240 + 17 Pi^8/161280 - 56 g[1] + 168 e g[1] - 280 e^2 g[1] + 280 e^3 g[1] - 168 e^4 g[1] + 56 e^5 g[1] - 8 e^6 g[1] - 140 g[1]^2 + 280 e g[1]^2 - 252 e^2 g[1]^2 + 112 e^3 g[1]^2 - 20 e^4 g[1]^2 - 56 g[1]^3 + 56 e g[1]^3 - 16 e^2 g[1]^3 - 2 g[1]^4 + 84 g[2] - 140 e g[2] + 140 e^2 g[2] - 84 e^3 g[2] + 28 e^4 g[2] - 4 e^5 g[2] + 140 g[1] g[2] - 168 e g[1] g[2] + 84 e^2 g[1] g[2] - 16 e^3 g[1] g[2] + 28 g[1]^2 g[2] - 12 e g[1]^2 g[2] - 21 g[2]^2 + 14 e g[2]^2 - 3 e^2 g[2]^2 - 2 g[1] g[2]^2 - 140 g[3]/3 + 140/3 e g[3] - 28 e^2 g[3] + 28/3 e^3 g[3] - 4/3 e^4 g[3] - 28 g[1] g[3] + 56/3 e g[1] g[3] - 4 e^2 g[1] g[3] - 4/3 g[1]^2 g[3] + 14/3 g[2] g[3] - 4/3 e g[2] g[3] - g[3]^2/9 + 35 g[4]/3 - 7 e g[4] + 7/3 e^2 g[4] - 1/3 e^3 g[4] + 7/3 g[1] g[4] - 2/3 e g[1] g[4] - 1/6 g[2] g[4] - 7 g[5]/5 + 7/15 e g[5] - 1/15 e^2 g[5] - 1/15 g[1] g[5] + 7 g[6]/90 - 1/90 e g[6] - g[7]/630 - 4 Log[4 Pi] - 49 Zeta[3] - 217 Zeta[5]/4 - 127 Zeta[7]/16], 10, 110][[1]]

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

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