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The Dirichlet series of the Möbius function is the reciprocal of the zeta Dirichlet series on Re s greater than 1

Statement

For $ℜ s > 1$ ,

$\sum_{n \geq 1} μ (n) n^{- s} = \frac{1}{ζ (s)},$

where $ζ (s) = \sum_{n \geq 1} n^{- s}$ .

Facts & Assumptions

Given: A complex number $s$ with $ℜ s > 1$ .

[L1]

The Möbius function satisfies $\sum_{d ∣ n} μ (d) = {\begin{cases} 1, & n = 1, \\ 0, & n > 1, \end{cases}$ that is, $μ * 1 = ε$ (Classical Möbius inversion over positive divisors, The Dirichlet-convolution identity and the constant-one function, The number-theoretic Möbius function $μ (n)$ from prime factorisation).

[L2]

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

[L3]

For every rational $q > 1$ , the series $\sum_{n \geq 1} n^{- q}$ converges (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ ).

Proof

technique · direct

1.1L1L3givenchoosealgebra

Write $σ : = ℜ s > 1$ and choose a rational $q$ with $1 < q < σ$ . Since $μ (n) \in {- 1, 0, 1}$ by [L1], one has $∣ μ (n) n^{- s} ∣ \leq n^{- σ} \leq n^{- q}, ∣ n^{- s} ∣ = n^{- σ} \leq n^{- q} .$ By [L3], both Dirichlet series therefore converge absolutely.

2.1L1L2step 1.1algebra

By [L2], $(\sum_{n \geq 1} μ (n) n^{- s}) (\sum_{n \geq 1} n^{- s}) = \sum_{n \geq 1} (μ * 1) (n) n^{- s} .$

3.1L1step 2.1∎

Step 2.1 and the identity in [L1] make the right-hand side equal to $1$ , so the first factor is the reciprocal of $ζ (s) = \sum_{n \geq 1} n^{- s}$ .

Depends on

Used by

Dependency tree · two levels

30 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 2.8 (standard reference, not scraped)
Jan-Hendrik Evertse, Analytic Number Theory, Chapter 2 (standard reference, not scraped)