In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Lambert series, named for
Johann Heinrich Lambert
Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subject ...
, is a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
taking the form
:
It can be resumed
formally by expanding the denominator:
:
where the coefficients of the new series are given by the
Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...
of ''a''
''n'' with the constant function 1(''n'') = 1:
:
This series may be inverted by means of the
Möbius inversion formula
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius.
A large gener ...
, and is an example of a
Möbius transform
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Pau ...
.
Examples
Since this last sum is a typical number-theoretic sum, almost any natural
multiplicative function
In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and
f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime.
An arithmetic function ''f''(''n'') is ...
will be exactly summable when used in a Lambert series. Thus, for example, one has
:
where
is the number of positive
divisors
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
of the number ''n''.
For the higher order
sum-of-divisor functions, one has
:
where
is any
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
and
:
is the divisor function. In particular, for
, the Lambert series one gets is
:
which is (up to the factor of
) the logarithmic derivative of the usual generating function for
partition number
In number theory, the partition function represents the number of possible partitions of a non-negative integer . For instance, because the integer 4 has the five partitions , , , , and .
No closed-form expression for the partition function is ...
s
:
Additional Lambert series related to the previous identity include those for the variants of the
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
given below
:
:
Related Lambert series over the
Moebius function
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Paul ...
include the following identities for any
prime
:
:
The proof of the first identity above follows from a multi-section (or bisection) identity of these
Lambert series generating functions in the following form where we denote
to be the Lambert series generating function of the arithmetic function ''f'':
:
The second identity in the previous equations follows from the fact that the coefficients of the left-hand-side sum are given by
:
where the function
is the multiplicative identity with respect to the operation of
Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...
of arithmetic functions.
For
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
:
:
For
Von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Definition
The von Mangold ...
:
:
For
Liouville's function The Liouville Lambda function, denoted by λ(''n'') and named after Joseph Liouville, is an important arithmetic function.
Its value is +1 if ''n'' is the product of an even number of prime numbers, and −1 if it is the product of an odd number of ...
:
:
with the sum on the right similar to the
Ramanujan theta function
In mathematics, particularly -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant fo ...
, or
Jacobi theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
. Note that Lambert series in which the ''a''
''n'' are
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s, for example, ''a''
''n'' = sin(2''n'' ''x''), can be evaluated by various combinations of the
logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula
\frac
where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f'' ...
s of Jacobi
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
s.
Generally speaking, we can extend the previous generating function expansion by letting
denote the characteristic function of the
powers,
, for positive natural numbers
and defining the generalized ''m''-Liouville lambda function to be the arithmetic function satisfying
. This definition of
clearly implies that
, which in turn shows that
:
We also have a slightly more generalized Lambert series expansion generating the
sum of squares function In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer as the sum of squares, where representations that differ only in the order of the summands or in the sign ...
in the form of
:
In general, if we write the Lambert series over
which generates the arithmetic functions
, the next pairs of functions correspond to other well-known convolutions expressed by their Lambert series generating functions in the forms of
:
where
is the multiplicative identity for
Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...
s,
is the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
for
powers,
denotes the characteristic function for the squares,
which counts the number of distinct prime factors of
(see
prime omega function In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. Thereby \omega(n) (little omega) counts each ''distinct'' prime factor, whereas the related function \Omega(n) (big omega) ...
),
is
Jordan's totient function Let k be a positive integer. In number theory, the Jordan's totient function J_k(n) of a positive integer n equals the number of k-tuples of positive integers that are less than or equal to n and that together with n form a coprime set of k+1 intege ...
, and
is the
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
(see
Dirichlet convolutions).
The conventional use of the letter ''q'' in the summations is a historical usage, referring to its origins in the theory of elliptic curves and theta functions, as the
nome.
Alternate form
Substituting
one obtains another common form for the series, as
:
where
:
as before. Examples of Lambert series in this form, with
, occur in expressions for the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
for odd integer values; see
Zeta constants for details.
Current usage
In the literature we find ''Lambert series'' applied to a wide variety of sums. For example, since
is a
polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...
function, we may refer to any sum of the form
:
as a Lambert series, assuming that the parameters are suitably restricted. Thus
:
which holds for all complex ''q'' not on the unit circle, would be considered a Lambert series identity. This identity follows in a straightforward fashion from some identities published by the Indian mathematician
S. Ramanujan. A very thorough exploration of Ramanujan's works can be found in the works by
Bruce Berndt.
Factorization theorems
A somewhat newer construction recently published over 2017–2018 relates to so-termed ''Lambert series factorization theorems'' of the form
:
where
is the respective sum or difference of the
restricted partition functions
which denote the number of
's in all partitions of
into an ''even'' (respectively, ''odd'') number of distinct parts. Let
denote the invertible lower triangular sequence whose first few values are shown in the table below.
Another characteristic form of the Lambert series factorization theorem expansions is given by
:
where
is the (infinite)
q-Pochhammer symbol. The invertible matrix products on the right-hand-side of the previous equation correspond to inverse matrix products whose lower triangular entries are given in terms of the
partition function and the
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
by the
divisor sums
:
The next table lists the first several rows of these corresponding inverse matrices.
We let
denote the sequence of interleaved
pentagonal numbers
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...
, i.e., so that the
pentagonal number theorem
In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that
:\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\right ...
is expanded in the form of
:
Then for any Lambert series
generating the sequence of
, we have the corresponding inversion relation of the factorization theorem expanded above given by
:
This work on Lambert series factorization theorems is extended in to more general expansions of the form
:
where
is any (partition-related) reciprocal generating function,
is any
arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
, and where the
modified coefficients are expanded by
:
The corresponding inverse matrices in the above expansion satisfy
:
so that as in the first variant of the Lambert factorization theorem above we obtain an inversion relation for the right-hand-side coefficients of the form
:
Recurrence relations
Within this section we define the following functions for natural numbers
:
:
:
We also adopt the notation from the
previous section that
:
where
is the infinite
q-Pochhammer symbol. Then we have the following recurrence relations for involving these functions and the
pentagonal numbers
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...
proved in:
:
:
Derivatives
Derivatives of a Lambert series can be obtained by differentiation of the series termwise with respect to
. We have the following identities for the termwise
derivatives of a Lambert series for any
:
:
where the bracketed triangular coefficients in the previous equations denote the
Stirling numbers of the first and second kinds.
We also have the next identity for extracting the individual coefficients of the terms implicit to the previous expansions given in the form of
:
Now if we define the functions
for any
by
:
where
denotes
Iverson's convention
In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement. ...
, then we have the coefficients for the
derivatives of a Lambert series
given by
:
Of course, by a typical argument purely by operations on formal power series we also have that
:
See also
*
Erdős–Borwein constant The Erdős–Borwein constant is the sum of the Reciprocal (mathematics), reciprocals of the Mersenne prime, Mersenne numbers. It is named after Paul Erdős and Peter Borwein.
By definition it is:
:E=\sum_^\frac\approx1.606695152415291763\dots
Eq ...
*
Arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
*
Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb are two arithmetic fun ...
References
*
*
*
*
*
* {{cite arXiv, last=Schmidt, first=Maxie Dion, date=2020-04-06, title=A catalog of interesting and useful Lambert series identities, class=math.NT, eprint=2004.02976
Analytic number theory
Q-analogs
Mathematical series