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Multiplying absolutely convergent Dirichlet series gives Dirichlet convolution
Statement
Let . On every half-plane where both Dirichlet series converge absolutely,
where is the Dirichlet convolution of Dirichlet convolution of arithmetic functions.
Facts & Assumptions
Given: Arithmetic functions and a point where both Dirichlet series converge absolutely.
Dirichlet convolution is (Dirichlet convolution of arithmetic functions).
A Dirichlet series is a series over positive integers (Dirichlet series).
Proof
Absolute convergence makes the double series absolutely convergent, so its terms may be regrouped by the product . The coefficient of in that regrouping is by [L1].
Therefore the product of the two Dirichlet series is the Dirichlet series of the convolution.
Depends on
Used by
- The Dirichlet series of Euler's totient is zeta of s minus 1 divided by zeta of s on Re s greater than 2 Corollary
- The Dirichlet series of the Möbius function is the reciprocal of the zeta Dirichlet series on Re s greater than 1 Corollary
- The divisor-counting Dirichlet series is the square of the zeta Dirichlet series on Re s greater than 1 Corollary
- The first coefficients of 1 over zeta are the Möbius values Example
- A multiplicative Dirichlet series factors as an Euler product on its absolute half-plane Theorem
- The logarithmic derivative of the zeta Dirichlet series is the Dirichlet series of the von Mangoldt function on Re s greater than 1 Theorem
Dependency tree · two levels
8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Leonard Tomczak, Analytic Number Theory, Theorem 3.2 (standard reference, not scraped)
- Kiran S. Kedlaya, Notes on Analytic Number Theory, Definition 2.5 (standard reference, not scraped)