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Dirichlet convolution of arithmetic functions
Definition
Let be arithmetic functions. Their Dirichlet convolution is the arithmetic function defined by
for each positive integer , where divisibility is that of Divisibility in : when for some integer .
This sum is finite: the positive divisors of form a finite interval in the divisibility poset by The divisibility poset is lower-finite, and each divisor interval factorises as a product of finite chains of prime exponents, and the summation is the finite commutative-monoid sum of A finite sum in a commutative monoid indexed by an arbitrary finite set.
Remarks
- The sum runs over positive divisors only. No value at occurs anywhere in the definition.
Depends on
- Arithmetic functions on the positive integers
- Divisibility in $\mathbb{Z}$: $d \mid a$ when $a = dq$ for some integer $q$
- A finite sum in a commutative monoid indexed by an arbitrary finite set
- The divisibility poset is lower-finite, and each divisor interval factorises as a product of finite chains of prime exponents
Used by
- A Dirichlet-convolution table through 12 Example
- The convolution 1*λ detects perfect squares Proposition
- The divisor functions arise by Dirichlet convolution Proposition
- An arithmetic function has a Dirichlet inverse exactly when its value at 1 is nonzero Theorem
- Arithmetic functions form a commutative ring under pointwise addition and Dirichlet convolution Theorem
- Dirichlet convolution preserves multiplicativity, and multiplicative inverses stay multiplicative Theorem
- The divisor sum of von Mangoldt is the arithmetic-function logarithm Theorem
Dependency tree · two levels
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Sources
- Victor Shoup, A Computational Introduction to Number Theory and Algebra, Section 2.9 (standard reference, not scraped)
- Kiran S. Kedlaya, An Introduction to Analytic Number Theory, Definition 3.5 (standard reference, not scraped)