A074760 - OEIS

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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

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OFFSET

0,2

COMMENTS

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

REFERENCES

H. M. Edwards, Riemann's Zeta Function, Dover Publications Inc. 1974, p. 160.

S. J. Patterson, "An introduction to the theory of the Riemann Zeta-function", Cambridge Studies in Advanced Mathematics 14, p. 34.

LINKS

Table of n, a(n) for n=0..103.

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.

FORMULA

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

EXAMPLE

0.023095708966121033814310247906495291621932127152050759525392...

MATHEMATICA

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