Keiper-Li coefficients

Symbol: KeiperLiLambda —

\lambda_{n}

— Keiper-Li coefficient

KeiperLiLambda(n), rendered as

\lambda_{n}

, denotes a power series coefficient associated with the Riemann zeta function.

The definition fcab61 follows Keiper (1992). Li (1997) defines the coefficients with a different scaling factor, equivalent to

n \lambda_{n}

in Keiper's (and Fungrim's) notation.

References:

https://doi.org/10.2307/2153215
https://doi.org/10.1006/jnth.1997.2137
https://doi.org/10.7169/facm/1317045228

Definitions:

Fungrim symbol	Notation	Short description
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient

Source code for this entry:

Entry(ID("432bee"),
    SymbolDefinition(KeiperLiLambda, KeiperLiLambda(n), "Keiper-Li coefficient"),
    Description(SourceForm(KeiperLiLambda(n)), ", rendered as", KeiperLiLambda(n), ", denotes a power series coefficient associated with the Riemann zeta function. "),
    Description("The definition", EntryReference("fcab61"), "follows Keiper (1992). Li (1997) defines the coefficients with a different scaling factor, equivalent to", Mul(n, KeiperLiLambda(n)), "in Keiper's (and Fungrim's) notation."),
    References("https://doi.org/10.2307/2153215", "https://doi.org/10.1006/jnth.1997.2137", "https://doi.org/10.7169/facm/1317045228"))

fcab61

\lambda_{n} = \frac{1}{n !} \left[ \frac{d^{n}}{{d s}^{n}} \log\!\left(2 \xi\!\left(\frac{s}{s - 1}\right)\right) \right]_{s = 0}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\lambda_{n} = \frac{1}{n !} \left[ \frac{d^{n}}{{d s}^{n}} \log\!\left(2 \xi\!\left(\frac{s}{s - 1}\right)\right) \right]_{s = 0}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`Factorial`	$n !$	Factorial
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Log`	$\log(z)$	Natural logarithm
`RiemannXi`	$\xi(s)$	Riemann xi-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("fcab61"),
    Formula(Equal(KeiperLiLambda(n), Mul(Div(1, Factorial(n)), ComplexDerivative(Log(Mul(2, RiemannXi(Div(s, Sub(s, 1))))), For(s, 0, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

cce75b

\lambda_{n} = \frac{1}{n} \sum_{\textstyle{k \in \mathbb{Z} \atop k \ne 0}} \left(1 - {\left(\frac{\rho_{k}}{\rho_{k} - 1}\right)}^{n}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\lambda_{n} = \frac{1}{n} \sum_{\textstyle{k \in \mathbb{Z} \atop k \ne 0}} \left(1 - {\left(\frac{\rho_{k}}{\rho_{k} - 1}\right)}^{n}\right)

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("cce75b"),
    Formula(Equal(KeiperLiLambda(n), Mul(Div(1, n), Sum(Parentheses(Sub(1, Pow(Div(RiemannZetaZero(k), Sub(RiemannZetaZero(k), 1)), n))), ForElement(k, ZZ), NotEqual(k, 0))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

081205

\lambda_{0} = 0

TeX:

\lambda_{0} = 0

Definitions:

Fungrim symbol	Notation	Short description
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient

Source code for this entry:

Entry(ID("081205"),
    Formula(Equal(KeiperLiLambda(0), 0)))

d8d820

\lambda_{1} = \frac{\gamma}{2} + 1 - \frac{\log\!\left(4 \pi\right)}{2}

TeX:

\lambda_{1} = \frac{\gamma}{2} + 1 - \frac{\log\!\left(4 \pi\right)}{2}

Definitions:

Fungrim symbol	Notation	Short description
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("d8d820"),
    Formula(Equal(KeiperLiLambda(1), Sub(Add(Div(ConstGamma, 2), 1), Div(Log(Mul(4, Pi)), 2)))))

faf448

Table of

\lambda_{n}

to 50 digits for

0 \le n \le 30

$n$	$\lambda_{n} \; (\text{nearest } 50 \text{D})$
0	0
1	0.023095708966121033814310247906495291621932127152051
2	0.046172867614023335192864243096033943387066108314123
3	0.069212973518108267930497348872601068994212026393200
4	0.092197619873060409647627872409439018065541673490213
5	0.11510854289223549048622128109857276671349132303596
6	0.13792766871372988290416713700341666356138966078654
7	0.16063715965299421294040287257385366292282442046163
8	0.18321945964338257908193931774721859848998098273432
9	0.20565733870917046170289387421343304741236553410044
10	0.22793393631931577436930340573684453380748385942738
11	0.25003280347456327821404973571398176484638012641151
12	0.27193794338538498733992383249265390667786600994911
13	0.29363385060368815285418215009889439246684857721098
14	0.31510554847718560800576009263276843951188505373007
15	0.33633862480178623056900742916909716316435743073656
16	0.35731926555429953996369166686545971896237127626351
17	0.37803428659512958242032593887899541131751543174423
18	0.39847116323842905329183170701797400318996274958010
19	0.41861805759536317393727500409965507879749928476235
20	0.43846384360466075647997306767236011141956127351910
21	0.45799812967347233249339981618322155418048244629837
22	0.47721127886067612259488922142809435229670372335579
23	0.49609442654413481917007764284527175942412402470703
24	0.51463949552297154237641907033478550942000739520745
25	0.53283920851566303199773431527093937195060386800653
26	0.55068709802460266427322142354105110711472096137358
27	0.56817751354772529773768642819955547787404863111699
28	0.58530562612777004271739082427374534543603950676386
29	0.60206743023974549532618306036749838157067445931420
30	0.61845974302711452077115586049317787034309975741269

Definitions:

Fungrim symbol	Notation	Short description
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient

Source code for this entry:

Entry(ID("faf448"),
    Description("Table of", KeiperLiLambda(n), "to 50 digits for", LessEqual(0, n, 30)),
    Table(Var(n), TableValueHeadings(n, NearestDecimal(KeiperLiLambda(n), 50)), TableSplit(1), List(Tuple(0, Decimal("0")), Tuple(1, Decimal("0.023095708966121033814310247906495291621932127152051")), Tuple(2, Decimal("0.046172867614023335192864243096033943387066108314123")), Tuple(3, Decimal("0.069212973518108267930497348872601068994212026393200")), Tuple(4, Decimal("0.092197619873060409647627872409439018065541673490213")), Tuple(5, Decimal("0.11510854289223549048622128109857276671349132303596")), Tuple(6, Decimal("0.13792766871372988290416713700341666356138966078654")), Tuple(7, Decimal("0.16063715965299421294040287257385366292282442046163")), Tuple(8, Decimal("0.18321945964338257908193931774721859848998098273432")), Tuple(9, Decimal("0.20565733870917046170289387421343304741236553410044")), Tuple(10, Decimal("0.22793393631931577436930340573684453380748385942738")), Tuple(11, Decimal("0.25003280347456327821404973571398176484638012641151")), Tuple(12, Decimal("0.27193794338538498733992383249265390667786600994911")), Tuple(13, Decimal("0.29363385060368815285418215009889439246684857721098")), Tuple(14, Decimal("0.31510554847718560800576009263276843951188505373007")), Tuple(15, Decimal("0.33633862480178623056900742916909716316435743073656")), Tuple(16, Decimal("0.35731926555429953996369166686545971896237127626351")), Tuple(17, Decimal("0.37803428659512958242032593887899541131751543174423")), Tuple(18, Decimal("0.39847116323842905329183170701797400318996274958010")), Tuple(19, Decimal("0.41861805759536317393727500409965507879749928476235")), Tuple(20, Decimal("0.43846384360466075647997306767236011141956127351910")), Tuple(21, Decimal("0.45799812967347233249339981618322155418048244629837")), Tuple(22, Decimal("0.47721127886067612259488922142809435229670372335579")), Tuple(23, Decimal("0.49609442654413481917007764284527175942412402470703")), Tuple(24, Decimal("0.51463949552297154237641907033478550942000739520745")), Tuple(25, Decimal("0.53283920851566303199773431527093937195060386800653")), Tuple(26, Decimal("0.55068709802460266427322142354105110711472096137358")), Tuple(27, Decimal("0.56817751354772529773768642819955547787404863111699")), Tuple(28, Decimal("0.58530562612777004271739082427374534543603950676386")), Tuple(29, Decimal("0.60206743023974549532618306036749838157067445931420")), Tuple(30, Decimal("0.61845974302711452077115586049317787034309975741269")))))

706f66

Table of

\lambda_{{10}^{n}}

to 50 digits for

0 \le n \le 5

$n$	$\lambda_{{10}^{n}} \; (\text{nearest } 50 \text{D})$
0	0.023095708966121033814310247906495291621932127152051
1	0.22793393631931577436930340573684453380748385942738
2	1.1860377537679132992736469839793792693298702359323
3	2.3260531616864664574065046940832238158044982041693
4	3.4736579732241613740180609478145593215167373519711
5	4.6258078240690223140941603808334320467617286152507

Definitions:

Fungrim symbol	Notation	Short description
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("706f66"),
    Description("Table of", KeiperLiLambda(Pow(10, n)), "to 50 digits for", LessEqual(0, n, 5)),
    Table(Var(n), TableValueHeadings(n, NearestDecimal(KeiperLiLambda(Pow(10, n)), 50)), TableSplit(1), List(Tuple(0, Decimal("0.023095708966121033814310247906495291621932127152051")), Tuple(1, Decimal("0.22793393631931577436930340573684453380748385942738")), Tuple(2, Decimal("1.1860377537679132992736469839793792693298702359323")), Tuple(3, Decimal("2.3260531616864664574065046940832238158044982041693")), Tuple(4, Decimal("3.4736579732241613740180609478145593215167373519711")), Tuple(5, Decimal("4.6258078240690223140941603808334320467617286152507")))))

64bd32

\left(\operatorname{RH}\right) \;\implies\; \left(\lambda_{n} \sim \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}, \; n \to \infty\right)

References:

https://doi.org/10.7169/facm/1317045228

TeX:

\left(\operatorname{RH}\right) \;\implies\; \left(\lambda_{n} \sim \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}, \; n \to \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`RiemannHypothesis`	$\operatorname{RH}$	Riemann hypothesis
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("64bd32"),
    Formula(Implies(RiemannHypothesis, AsymptoticTo(KeiperLiLambda(n), Sub(Div(Log(n), 2), Div(Sub(Add(Log(Mul(2, Pi)), 1), ConstGamma), 2)), n, Infinity))),
    References("https://doi.org/10.7169/facm/1317045228"))

e68f82

\left(\operatorname{RH}\right) \iff \left(\lambda_{n} > 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right)

References:

https://doi.org/10.1006/jnth.1997.2137

TeX:

\left(\operatorname{RH}\right) \iff \left(\lambda_{n} > 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right)

Definitions:

Fungrim symbol	Notation	Short description
`RiemannHypothesis`	$\operatorname{RH}$	Riemann hypothesis
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("e68f82"),
    Formula(Equivalent(RiemannHypothesis, All(Greater(KeiperLiLambda(n), 0), ForElement(n, ZZGreaterEqual(1))))),
    References("https://doi.org/10.1006/jnth.1997.2137"))

a5d65f

\left(\operatorname{RH}\right) \iff \left(\sum_{n=1}^{\infty} {\left|\lambda_{n} - a(n)\right|}^{2} < \infty\; \text{ where } a(n) = \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}\right)

References:

https://doi.org/10.7169/facm/1317045228

TeX:

\left(\operatorname{RH}\right) \iff \left(\sum_{n=1}^{\infty} {\left|\lambda_{n} - a(n)\right|}^{2} < \infty\; \text{ where } a(n) = \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}\right)

Definitions:

Fungrim symbol	Notation	Short description
`RiemannHypothesis`	$\operatorname{RH}$	Riemann hypothesis
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Abs`	$\left\|z\right\|$	Absolute value
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`Infinity`	$\infty$	Positive infinity
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstGamma`	$\gamma$	The constant gamma (0.577...)

Source code for this entry:

Entry(ID("a5d65f"),
    Formula(Equivalent(RiemannHypothesis, Where(Less(Sum(Pow(Abs(Sub(KeiperLiLambda(n), a(n))), 2), For(n, 1, Infinity)), Infinity), Equal(a(n), Sub(Div(Log(n), 2), Div(Sub(Add(Log(Mul(2, Pi)), 1), ConstGamma), 2)))))),
    References("https://doi.org/10.7169/facm/1317045228"))

8f8fb7

\left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1} \text{ with } \operatorname{Im}\!\left(\rho_{n}\right) < T\right) \;\implies\; \left(\lambda_{n} \ge 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 0} \text{ with } n \le {T}^{2}\right)

Assumptions:

T \in \left[0, \infty\right)

References:

https://arxiv.org/abs/1703.02844

TeX:

\left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1} \text{ with } \operatorname{Im}\!\left(\rho_{n}\right) < T\right) \;\implies\; \left(\lambda_{n} \ge 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 0} \text{ with } n \le {T}^{2}\right)

T \in \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Re`	$\operatorname{Re}(z)$	Real part
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Im`	$\operatorname{Im}(z)$	Imaginary part
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`Pow`	${a}^{b}$	Power
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("8f8fb7"),
    Formula(Implies(All(Equal(Re(RiemannZetaZero(n)), Div(1, 2)), ForElement(n, ZZGreaterEqual(1)), Less(Im(RiemannZetaZero(n)), T)), All(GreaterEqual(KeiperLiLambda(n), 0), ForElement(n, ZZGreaterEqual(0)), LessEqual(n, Pow(T, 2))))),
    Variables(T),
    Assumptions(Element(T, ClosedOpenInterval(0, Infinity))),
    References("https://arxiv.org/abs/1703.02844"))

Definitions

Representations

Specific values

Asymptotics

Riemann hypothesis (Li's criterion)