In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory. This was developed by Johann Peter Gustav Lejeune Dirichlet, a German mathematician. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Peter Gustav Lejeune Dirichlet. ...

If f and g are two arithmetic functions (i.e. functions from the positive integers to the complex numbers), one defines a new arithmetic function f * g, the Dirichlet convolution of f and g, by The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...

where the sum extends over all positive divisors d of n. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

Some general properties of this operation include:

• If both f and g are multiplicative, then so is f * g. (Note however that the convolution of two completely multiplicative functions need not be completely multiplicative.)
• f * g = g * f (commutativity)
• (f * g) * h = f * (g * h) (associativity)
• f * (g + h) = f * g + f * h (distributivity)
• f * ε = ε * f = f, where ε is the function defined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1.
• For each f for which f(1) ≠ 0 there exists a g such that f * g = ε. g is called the Dirichlet inverse of f.
• In particular, every multiplicative f has a Dirichlet inverse g that is also multiplicative.

With addition and Dirichlet convolution, the set of arithmetic functions forms a commutative ring with multiplicative identity ε, the Dirichlet ring (note that it is not a field because some arithmetic functions do not have Dirichlet inverses). The units of this ring are the arithmetical functions f with f(1) ≠ 0. 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). ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

Furthermore, the multiplicative functions with convolution form an abelian group with identity element ε. The article on multiplicative functions lists several convolution relations among important multiplicative functions. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... 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). ...

If f is an arithmetic function, one defines its Dirichlet series generating function by In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

for those complex arguments s for which the series converges (if there are any). The multiplication of Dirichlet series is compatible with Dirichlet convolution in the following sense: In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...

for all s for which the left hand side exists. This is akin to the convolution theorem if one thinks of Dirichlet series as a Fourier transform. In mathematics, the convolution theorem states that the Fourier transform of a convolution is the point-wise product of Fourier transforms. ... The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...