Math 2.0

Number theory Proposition

Dirichlet convolution formula

Builds on Dirichlet convolution

Statement

Let ff and gg be arithmetic functions, and let nn be a positive integer. Then (f∗g)(n)=∑a,b≥1ab=nf(a) g(b),(f * g)(n) = \sum_{\substack{a, b \ge 1 \\ ab = n}} f(a) \, g(b), where the sum runs over the ordered pairs of positive integers (a,b)(a, b) with ab=nab = n, and ∗* is the Dirichlet convolution.

Remarks

The map d↦(d,n/d)d \mapsto (d, n/d) is a bijection from the divisors of nn to these pairs. This gives the formula. The symmetric form shows at once that f∗g=g∗ff * g = g * f. For example, (f∗g)(6)=f(1)g(6)+f(2)g(3)+f(3)g(2)+f(6)g(1)(f * g)(6) = f(1)g(6) + f(2)g(3) + f(3)g(2) + f(6)g(1).

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Arithmetic function, definition, no evidence levelArithmetic functiondefinitionDivisibility, definition, no evidence levelDivisibilitydefinitionDivisors of a natural number, definition, no evidence levelDivisors of a natural…definitionDirichlet convolution, definition, no evidence levelDirichlet convolutiondefinitionDirichlet convolution formula, statement, no evidence levelDirichlet convolution…statement
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  1. Arithmetic function definition
  2. Divisibility definition
  3. Divisors of a natural number definition
  4. Dirichlet convolution definition
  5. Dirichlet convolution formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dirichlet_convolution, Definition
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