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A multiplicative Dirichlet series factors as an Euler product on its absolute half-plane

Statement

Let $f$ be a multiplicative arithmetic function. If

$\sum_{n \geq 1} ∣ f (n) ∣ n^{- σ} < \infty,$

then for every $s$ with $ℜ s \geq σ$ ,

$\sum_{n \geq 1} f (n) n^{- s} = \prod_{p} \sum_{k \geq 0} f (p^{k}) p^{- k s},$

where the infinite product is the limit of the finite prime products.

Facts & Assumptions

Given: A multiplicative arithmetic function $f$ and a complex number $s$ with $ℜ s \geq σ$ .

[L1]

Multiplicative functions satisfy $f (m n) = f (m) f (n)$ for coprime $m, n$ (Multiplicative arithmetic functions).

[L2]

Every positive integer has a unique prime factorization (The fundamental theorem of arithmetic: every integer $n \geq 1$ is a product of primes, and the factorisation is unique up to order — if $\prod_{i < r} p_{i} = \prod_{j < s} q_{j}$ with every $p_{i}$ and $q_{j}$ prime, then $r = s$ and $q_{i} = p_{π (i)}$ for some $π \in Sym (r)$ ).

[L3]

Products of absolutely convergent Dirichlet series multiply by convolution (Multiplying absolutely convergent Dirichlet series gives Dirichlet convolution).

Proof

technique · direct

1.1

For a finite set $P$ of primes, expand $\prod_{p \in P} \sum_{k \geq 0} f (p^{k}) p^{- k s} .$ Using multiplicativity [L1] and unique factorization [L2], this is exactly $\sum_{\begin{array}{c} n \geq 1 \\ all prime factors of n lie in P \end{array}} f (n) n^{- s} .$

L1L2L3givenalgebra

2.1

As $P$ increases, these partial Euler products exhaust the original Dirichlet series. Because $ℜ s \geq σ$ and the series $\sum ∣ f (n) ∣ n^{- σ}$ converges, the omitted tail tends to $0$ absolutely. Hence the finite prime products converge to $\sum_{n \geq 1} f (n) n^{- s}$ .

step 1.1givenalgebra∎

Depends on

Used by

Dependency tree · two levels

31 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

Kiran S. Kedlaya, Notes on Analytic Number Theory, Definition 2.6 (standard reference, not scraped)
Leonard Tomczak, Analytic Number Theory, Theorem 3.2 (standard reference, not scraped)