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Keiper-Li coefficients
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Numbers
n 
λn
INPUT{numbers.yaml} (not shown in preview)
Definition
The Keiper-Li coefficients λn, n0, are defined as λn=ρ(1(11ρ)n), where the sum is interpreted as limTρ:|ρ|T over the set of non-trivial zeros ρ of the Riemann zeta function.
Parameters
n
—   integer (n0)
Formulas
(1)
λn=λn, if we extend the same definition to negative indices.
(2)
λn=1(n1)!dndsn(sn1logξ(s))|s=1 for n1 and λ0=0, where ξ(s) is the Riemann ξ function ξ(s)=12s(s1)πs/2Γ(s/2)ζ(s)
(3)
log(2ξ(1/z))=n=1λnn(1z)n, which is how Keiper originally defined the constants λnKeiper=λn/n.
(4)
ddzlogξ(11z)=n=0λn+1zn.
(5)
λ1=1+γ/2log212logπ, where γ is the Euler-Mascheroni constant.
(6)
λ2=1+γγ2+π2/8 2log2logπ2γ1, where γ1 is the first Stieltjes constant.
(7)
λ3=12(2+34π26log23logπ12γ1+γ(3+2(γ3)γ+6γ1)+3γ274ζ(3)).
(8)
λn+1=λn+1n!dndsn(snξ(s)ξ(s))|s=1 for n0.
Comments
(9)
Keiper [5] originally defined λn with a different normalization: λnKeiper=1nλn.
(10)
The Riemann hypothesis is equivalent to λn0 for all n. This is Li's criterion.
References
[1]
Juan Arias de Reyna, "Asymptotics of Keiper-Li coefficients", Functiones et Approximatio Commentarii Mathematici. 45(1) (2011), 7-21. (doi)
[2]
M.W. Coffey, "Relations and positivity results for the derivatives of the Riemann ξ function", J. Comput. Appl. Math. 166 (2004), 525-534.
[3]
Donal F. Connon, "A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants", (2009), 50 pages.
[4]
Fredrik Johansson, "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives", Numerical Algorithms. 69(2) (2015), 253-270. (arXiv) (doi)
[5]
J. B. Keiper, "Power series expansions of Riemann's ξ function", Math. Comp. 58 (1992), 765-773. (doi)
[6]
Jeffrey C. Lagarias, "Li coefficients for automorphic L-functions", Ann. Inst. Fourier, Grenoble, Tome 57, no 5 (2007), 1689-1740. http://aif.cedram.org/item?id=AIF_2007__57_5_1689_0
[7]
Xian-Jin Li, "The Positivity of a Sequence of Numbers and the Riemann Hypothesis", Journal of Number Theory 65 (1997), 325-333.
Links
Data properties
Entries are of type: real number
Table is complete: no