Li's criterion states that the Riemann hypothesis is equivalent to the statement that, for
(1)
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where is the xi-function, for every positive integer (Li 1997). Li's constants can be written in alternate form as
(2)
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(Coffey 2004).
can also be written as a sum of nontrivial zeros of as
(3)
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(Li 1997, Coffey 2004).
A recurrence for in terms of is given by
(4)
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(Coffey 2004).
The first few explicit values of the constantes are
(5)
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(6)
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(7)
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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
(8)
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where
(9)
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and .
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
(10)
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(left figure above), larger terms show clear variation from a parabolic fit (right figure).