In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... For other persons named Eric Bell, see Eric Bell (disambiguation). ...

Given an arithmetic function f and a prime p, define the formal power series f_p(x), called the Bell series of f modulo p as In number theory and computability theory, subfields of mathematics, a number-theoretic function is any function whose domain is the set of natural numbers. ...

Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem. Given multiplicative functions f and g, one has f = g if and only if 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). ...

f_p(x) = g_p(x) for all primes p.

Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions f and g, let h = f * g be their Dirichlet convolution. Then for every prime p, one has In number theory and computability theory, subfields of mathematics, a number-theoretic function is any function whose domain is the set of natural numbers. ... In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory. ...

In particular, this makes it trivial to find the Bell series of a Dirichlet inverse. In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory. ...

If f is completely multiplicative, then 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). ...

Examples

The following is a table of the Bell series of well-known arithmetic functions.

• The Moebius function μ has μ_p(x) = 1 − x.
• Euler's Totient φ has
• The identity function I has I_p(x) = 1.
• The Liouville function λ has
• The power function Id_k has Here, Id_k is the completely multiplicative function .
• The divisor function σ_k has

The classical Möbius function is an important multiplicative function in number theory and combinatorics. ... In number theory, the totient φ(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. ... The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory. ... 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...

References

• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9