A195423 -id:A195423 - 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

A245275 Decimal expansion of sum_{r in Z}(1/r^2) where Z is the set of all nontrivial zeros r of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 16 2014

			-0.046154317295804602757107990379077303530267962324144990348848453508...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.21 Stieltjes Constants, p. 168.

Eric Weisstein's MathWorld, Stieltjes Constants
Wikipedia, Stieltjes constants

Cf. A082633, A074760, A195423, A245276.

Join[{0}, RealDigits[-Pi^2/8 + EulerGamma^2 + 2*StieltjesGamma[1] + 1, 10, 104] // First]

-Pi^2/8+Euler^2+1+2*intnum(x=0,oo,(1/tanh(Pi*x)-1)*(x*log(1+x^2)-2*atan(x))/(2*(1+x^2))) \\ Charles R Greathouse IV, Mar 10 2016

-Pi^2/8 + gamma^2 + 2*gamma(1) + 1, where gamma is Euler's constant and gamma(1) is the first Stieltjes constant.

A245276 Decimal expansion of sum_{r in Z}(1/r^3) where Z is the set of all nontrivial zeros r of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 16 2014

			-0.000111158231452105922762668238914578473964189248986518770273452672891213...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.21 Stieltjes Constants, p. 168.

Eric Weisstein's MathWorld, Stieltjes Constants
Wikipedia, Stieltjes constants

Cf. A074760, A195423, A245275.

Join[{0, 0, 0}, RealDigits[-7*Zeta[3]/8 + EulerGamma^3 + 3*EulerGamma*StieltjesGamma[1] + 3/2*StieltjesGamma[2] + 1, 10, 103] // First]

-7*zeta(3)/8 + gamma^3 + 3*gamma*gamma(1) + 3/2*gamma(2) + 1, where gamma is Euler's constant and gamma(n) is the n-th Stieltjes constant.

A233825 Decimal expansion of Nicolas's constant in his condition for the Riemann Hypothesis (RH).

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Dec 19 2013

Nicolas proved that RH is true if and only if limsup_{n-->infinity} (n/phi(n) - e^gamma*log(log(n)))*sqrt(log(n)) = e^gamma*(4 + gamma - log(4*Pi)), where phi(n) = A000010(n).

			3.64441509640737014106511619283514816005226024664324245685246375826374...

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. A.M.S., 50 (2013), 527-628; see p. 574.
Jean-Louis Nicolas, Small values of the Euler function and the Riemann hypothesis, Acta Arith., Vol. 155, No. 3 (2012), pp. 311-321; arXiv preprint, arXiv:1202.0729 [math.NT], 2012.

Cf. A000010, A001620, A195423, A216868, A218245.

RealDigits[Exp[EulerGamma]*(4 + EulerGamma - Log[4*Pi]), 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)

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

Equals e^gamma*(4 + gamma - log(4*Pi)), where gamma is the Euler-Mascheroni constant.

Equals e^gamma*(2 + beta), where beta = Sum 1/(rho*(1-rho)), where rho runs over all nonreal zeros of the zeta function.

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

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A245275 Decimal expansion of sum_{r in Z}(1/r^2) where Z is the set of all nontrivial zeros r of the Riemann zeta function.

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A245276 Decimal expansion of sum_{r in Z}(1/r^3) where Z is the set of all nontrivial zeros r of the Riemann zeta function.

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A233825 Decimal expansion of Nicolas's constant in his condition for the Riemann Hypothesis (RH).

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