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 |
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poincaré series