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A074760 Decimal expansion of lambda(1) in Li's criterion. 22
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 (list; constant; graph; refs; listen; history; text; internal format)
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.

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]]

PROG

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

CROSSREFS

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).

Sequence in context: A276994 A020823 A021437 * A137914 A098989 A175315

Adjacent sequences:  A074757 A074758 A074759 * A074761 A074762 A074763

KEYWORD

cons,nonn

AUTHOR

Benoit Cloitre, Sep 28 2002

EXTENSIONS

Name simplified by Eric W. Weisstein, Feb 08 2019

STATUS

approved

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Last modified May 29 07:49 EDT 2022. Contains 354122 sequences. (Running on oeis4.)