In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Johann Heinrich Lambert Johann Heinrich Lambert (August 26, 1728 – September 25, 1777), was a mathematician, physicist and astronomer. ... In mathematics, a series is often represented as the sum of a sequence of terms. ...

It can be resummed formally by expanding the denominator:

where the coefficients of the new series are given by the Dirichlet convolution of a_n with the constant function 1(n) = 1: In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory. ...

bm = (a * 1)(m) =	∑	an
	n | m

This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform. The classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. ... The Möbius transform should not be confused with Möbius transformations. ...

Examples

Since this last sum is a typical number-theoretic sum, almost any multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...

where σ₀(n) = d(n) is the number of positive divisors of the number n. Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is...

For the higher order sigma functions, one has Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is...

where α is any complex number and In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

σα(n) = (Idα * 1)(n) =	∑	dα
	d | n

is the divisor function.

Lambert series in which the a_n are trigonometric functions, for example, a_n=sin (2n x), can be evaluated by various combinations of the logarithmic derivatives of Jacobi theta functions. All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other... In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula f′/f where f′ is the derivative of f. ... In mathematics, theta functions are special functions of several complex variables. ...

Other Lambert series include those for the Mobius function μ(n): The classical Möbius function is an important multiplicative function in number theory and combinatorics. ...

For Euler's totient function φ(n): The first thousand values of φ(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...

For Liouville's function λ(n): The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory. ...

with the sum on the left similar to the Ramanujan theta function. In mathematics, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. ...

Alternate form

Substituting q = e ^{− z} one obtains another common form for the series, as

where

bm = (a * 1)(m) =	∑	an
	n | m

as before. Examples of Lambert series in this form, with z = 2π, occur in expressions for the Riemann zeta function for odd integer values; see Zeta constants for details. In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ...

See also

• Erdős–Borwein constant

The Erdős–Borwein constant is the sum of the reciprocals of the Mersenne numbers. ...

References

• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9
• Eric W. Weisstein, Lambert Series at MathWorld.