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Date: 25-4-2019
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Date: 30-3-2019
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Li's criterion states that the Riemann hypothesis is equivalent to the statement that, for
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(1) |
where is the xi-function,
for every positive integer
(Li 1997). Li's constants can be written in alternate form as
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(2) |
(Coffey 2004).
can also be written as a sum of nontrivial zeros
of
as
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(3) |
(Li 1997, Coffey 2004).
A recurrence for in terms of
is given by
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(4) |
(Coffey 2004).
The first few explicit values of the constantes are
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(5) |
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(6) |
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(7) |
where is the Euler-Mascheroni constant and
are Stieltjes constants.
can be computed efficiently in closed form using recurrence formulas due to Coffey (2004), namely
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(8) |
where
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(9) |
and .
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OEIS |
1 | 0.0230957... | A074760 |
2 | 0.0923457... | A104539 |
3 | 0.2076389... | A104540 |
4 | 0.3687904... | A104541 |
6 | 0.5755427... | A104542 |
7 | 1.1244601... | A306340 |
8 | 1.4657556... | A306341 |
Edwards 2001 (p. 160) gave a numerical value for , and numerical values to six digits up to
were tabulated by Coffey (2004).
While the values of up to
are remarkably well fit by a parabola with
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(10) |
(left figure above), larger terms show clear variation from a parabolic fit (right figure).
REFERENCES:
Bombieri, E. and Lagarias, J. C. "Complements to Li's Criterion for the Riemann Hypothesis." J. Number Th. 77, 274-287, 1999.
Coffey, M. W. "Relations and Positivity Results for Derivatives of the Riemann Function." J. Comput. Appl. Math. 166, 525-534, 2004.
Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.
Keiper, J. B. "Power Series Expansions of Riemann's Function." Math. Comput. 58, 765-773, 1992.
Li, X.-J. "The Positivity of a Sequence of Numbers and the Riemann Hypothesis." J. Number Th. 65, 325-333, 1997.
Sloane, N. J. A. Sequences A074760, A104539, A104540, A104541, and A104542 in "The On-Line Encyclopedia of Integer Sequences."
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