Lambert series
E279122
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lambert series canonical | 1 |
| Poincaré series | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2566432 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Lambert series Context triple: [Johann Heinrich Lambert, knownFor, Lambert series]
-
A.
Hahn series
Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.
-
B.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
C.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
D.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
E.
Taylor series
A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Lambert series Target entity description: Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
-
A.
Hahn series
Hahn series are formal power series with exponents in an ordered abelian group and well-ordered supports, providing a general framework for constructing large ordered fields that include structures like the surreal numbers.
-
B.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
C.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
D.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
E.
Taylor series
A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in mathematical analysis
ⓘ
concept in number theory ⓘ mathematical series ⓘ |
| appearsInWorkOf |
G. H. Hardy
ⓘ
Hans Rademacher ⓘ Srinivasa Ramanujan ⓘ |
| belongsTo | q-series in the theory of special functions ⓘ |
| canEncode |
Euler’s totient function φ(n)
ⓘ
surface form:
Euler totient function
Möbius function ⓘ divisor functions ⓘ partition function ⓘ |
| convergesFor | |q|<1 in many standard cases ⓘ |
| field |
mathematical analysis
ⓘ
number theory ⓘ |
| hasAlternativeDescription | power series whose coefficients are Dirichlet convolutions of arithmetic functions ⓘ |
| hasAnalyticAspect | studied via complex analysis of q in the unit disk ⓘ |
| hasCombinatorialAspect | interpreted as weighted counts of divisors ⓘ |
| hasExample |
\sum_{n=1}^{\infty} \frac{n q^n}{1-q^n} = \sum_{n=1}^{\infty} \sigma_1(n) q^n
ⓘ
\sum_{n=1}^{\infty} \frac{q^n}{1-q^n} = \sum_{n=1}^{\infty} d(n) q^n ⓘ \sum_{n=1}^{\infty} \mu(n) \frac{q^n}{1-q^n} ⓘ |
| hasGeneralForm | \sum_{n=1}^{\infty} a(n) \frac{q^n}{1-q^n} ⓘ |
| hasTransformation | can be inverted under suitable conditions to recover the underlying arithmetic function ⓘ |
| introducedBy | Johann Heinrich Lambert ⓘ |
| involves |
arithmetic functions
ⓘ
powers of a variable ⓘ |
| namedAfter | Johann Heinrich Lambert ⓘ |
| property |
can be transformed using modular transformations in suitable cases
ⓘ
often appear in identities involving divisor sums ⓘ often express arithmetic functions as coefficients of power series in q ⓘ |
| relatedConcept |
Lambert W function (later named in his honor)
ⓘ
surface form:
Lambert W function (distinct but historically related name)
|
| relatedTo |
Dirichlet series
ⓘ
Euler products ⓘ generating functions ⓘ mock theta functions ⓘ modular forms ⓘ theta functions ⓘ |
| specialCase | q-series ⓘ |
| typicalConstraintOnCoefficientFunction | a(n) is often multiplicative in number-theoretic applications ⓘ |
| usedFor |
deriving congruences for partition functions
ⓘ
expressing generating functions of divisor-type sequences ⓘ studying growth of arithmetic functions ⓘ |
| usedIn |
combinatorics
ⓘ
multiplicative number theory ⓘ partition theory ⓘ q-series ⓘ theory of modular forms ⓘ |
| variableUsuallyDenotedBy | q ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Lambert series Description of subject: Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poincaré series