21.11.01 · number-theory / dirichlet-series-arithmetic-functions

Arithmetic Functions, Dirichlet Convolution, and Möbius Inversion

shipped3 tiersLean: none

Anchor (Master): Apostol 1976 *Introduction to Analytic Number Theory* Ch. 2-3 (the Dirichlet-convolution ring, Bell series, averages); Möbius 1832 *Crelle* 9 (originator of the inversion function); Dedekind 1857 and Liouville 1857 (early systematic use of the convolution and multiplicativity); McCarthy 1986 *Introduction to Arithmetical Functions* (Springer Universitext); Tóth 2014 survey *Multiplicative arithmetic functions of several variables*

Intuition Beginner

Number theory is full of functions that read off some feature of a whole number from its factorisation. How many divisors does $n$ have? What is the sum of those divisors? How many numbers below $n$ share no common factor with it? Each of these is an arithmetic function — a rule that takes a positive whole number and returns a number. The number $12$ has six divisors ( $1, 2, 3, 4, 6, 12$ ); their sum is $28$ ; and exactly four numbers below $12$ ( $1, 5, 7, 11$ ) share no factor with it. Three different functions, all reading $12$ .

The surprise is that these functions are not isolated. There is a single multiplication rule that ties them together. To combine two arithmetic functions $f$ and $g$ at the number $n$ , you run over every way of splitting $n$ into an ordered pair of factors $d$ and $n / d$ , multiply $f (d)$ by $g (n / d)$ , and add up the results. This combining rule is called the Dirichlet convolution, written $f * g$ . It behaves like ordinary multiplication: you can reorder, regroup, and there is a "do-nothing" function that leaves everything unchanged.

One special function, the Möbius function $μ$ , acts as an undo button. If you build a function $g$ by summing another function $f$ over divisors, the Möbius function lets you recover $f$ from $g$ . This recovery trick — Möbius inversion — turns hard sums into easy ones and is the reason the Möbius function appears throughout number theory.

Visual Beginner

A diagram showing the number $12$ at the centre with its six divisors branching out, paired so that each divisor $d$ sits opposite its partner $12/ d$ : $1$ with $12$ , $2$ with $6$ , $3$ with $4$ . Each pairing is one term in a Dirichlet convolution. Beside it, a small table lists the Möbius function on the first several numbers, showing the pattern of $+ 1$ , $- 1$ , and $0$ .

$n$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$μ (n)$	$1$	$- 1$	$- 1$	$0$	$- 1$	$1$	$- 1$	$0$	$0$	$1$	$- 1$	$0$

The value $μ (n)$ is $+ 1$ when $n$ is a product of an even number of distinct primes (and for $n = 1$ ), $- 1$ for an odd number of distinct primes, and $0$ whenever a prime is repeated. The zeros at $4, 8, 9, 12$ mark the numbers divisible by a square.

Worked example Beginner

Build the divisor-counting function as a convolution, and check it at $n = 12$ .

Let $u$ be the function that is equal to $1$ at every number. The convolution $u * u$ at $n$ runs over every factor pair $d$ and $n / d$ , multiplies $u (d) = 1$ by $u (n / d) = 1$ , and adds. Each pair contributes $1$ , so $u * u$ at $n$ simply counts the divisors of $n$ . This counting function is called $d (n)$ .

Step 1. List the divisors of $12$ : they are $1, 2, 3, 4, 6, 12$ . There are six of them, so $d (12) = 6$ .

Step 2. Check this against the convolution. The factor pairs of $12$ are $(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1)$ — six ordered pairs. Each contributes $1 \times 1 = 1$ . The total is $6$ , matching $d (12) = 6$ .

Step 3. Try a prime power for contrast. For $8 = 2^{3}$ the divisors are $1, 2, 4, 8$ , four of them, so $d (8) = 4$ . A prime power $p^{k}$ always has $k + 1$ divisors, since the exponent can be any value from $0$ to $k$ .

What this tells us: a familiar counting function is secretly the convolution of the simplest function (the constant $1$ ) with itself. Recognising functions as convolutions is what lets a single algebraic rule organise the whole subject.

Check your understanding Beginner

Formal definition Intermediate+

An arithmetic function is a function $f : Z_{> 0} \to C$ . The set of all arithmetic functions is denoted $A$ . Following Apostol ^{[Apostol Ch. 2]}, $A$ carries an algebraic structure richer than pointwise operations.

Definition (Dirichlet convolution). For $f, g \in A$ , the Dirichlet convolution (or Dirichlet product) $f * g \in A$ is $$ (f * g)(n) = \sum_{d \mid n} f(d) , g!\left(\frac{n}{d}\right) = \sum_{\substack{ab = n \ a, b \geq 1}} f(a) , g(b), $$ the sum ranging over the ordered factorisations $n = ab$ into positive integers.

Definition (distinguished functions). Several arithmetic functions recur throughout the theory.

The identity for convolution $ε$ is $ε (1) = 1$ and $ε (n) = 0$ for $n > 1$ .
The unit function $1$ is $1 (n) = 1$ for all $n$ .
The power functions $N^{k}$ are $N^{k} (n) = n^{k}$ ; in particular $N^{0} = 1$ and $N = N^{1}$ is $N (n) = n$ .
The Möbius function $μ$ is $μ (1) = 1$ , $μ (n) = (- 1)^{k}$ if $n = p_{1} \dots p_{k}$ is a product of $k$ distinct primes, and $μ (n) = 0$ if $n$ is divisible by the square of a prime.
The Euler totient $φ (n) = # {1 \leq m \leq n : g cd (m, n) = 1}$ .
The divisor functions $σ_{k} = N^{k} * 1$ , so $σ_{k} (n) = \sum_{d ∣ n} d^{k}$ ; the special cases $d = σ_{0} = 1 * 1$ (number of divisors) and $σ = σ_{1}$ (sum of divisors).
The von Mangoldt function $Λ (n) = lo g p$ if $n = p^{m}$ is a prime power ( $m \geq 1$ ) and $Λ (n) = 0$ otherwise.
The Liouville function $λ (n) = (- 1)^{Ω (n)}$ , where $Ω (n)$ counts prime factors with multiplicity.

Definition (multiplicative and completely multiplicative). An arithmetic function $f$ with $f (1) = 1$ is multiplicative if $f (mn) = f (m) f (n)$ whenever $g cd (m, n) = 1$ , and completely multiplicative if $f (mn) = f (m) f (n)$ for all $m, n$ . The functions $μ, φ, σ_{k}, d$ are multiplicative; $1, N^{k}, λ, ε$ are completely multiplicative; $Λ$ is neither.

The triple $(A, +, *)$ — pointwise addition and Dirichlet convolution — is a commutative ring with multiplicative identity $ε$ . The notation $lo g$ denotes the arithmetic function $n \mapsto lo g n$ , an element of $A$ (registered in _meta/NOTATION.md).

Counterexamples to common slips

" $μ$ is completely multiplicative." It is multiplicative but not completely so: $μ (4) = 0 \neq = μ (2) μ (2) = 1$ . Multiplicativity only constrains coprime arguments, and $2$ is not coprime to itself. Completely multiplicative functions are determined by their values on primes alone; multiplicative ones need values on all prime powers.
" $σ$ counts divisors." The divisor count is $d = σ_{0}$ ; the divisor sum is $σ = σ_{1}$ . At $n = 6$ : $d (6) = 4$ but $σ (6) = 1 + 2 + 3 + 6 = 12$ .
"The convolution sums $f (d) g (d)$ ." The second argument is the complementary divisor $n / d$ , not $d$ . The operation $\sum_{d ∣ n} f (d) g (d)$ is not the Dirichlet convolution and is not associative in the same way.
" $Λ$ is multiplicative." It is not: $Λ (1) = 0 \neq = 1$ , so it fails even the normalisation $f (1) = 1$ that every multiplicative function obeys. Its role is via the identity $Λ = μ * lo g$ , not through multiplicativity.

Key theorem with proof Intermediate+

The signature theorem is the Möbius inversion formula, which expresses the invertibility of the unit function $1$ under Dirichlet convolution. Its content is that $μ$ is the Dirichlet inverse of $1$ .

Theorem (Möbius inversion). *In the ring $(A, +, *)$ the Möbius function $μ$ is the convolution inverse of the unit function $1$ :* $$ \mu * \mathbf{1} = \varepsilon, \qquad \text{i.e.} \qquad \sum_{d \mid n} \mu(d) = \begin{cases} 1 & n = 1, \ 0 & n > 1. \end{cases} $$ Consequently, for arithmetic functions $f, g$ , $$ g = f * \mathbf{1} \quad \Longleftrightarrow \quad f = g * \mu, $$ that is, $g (n) = \sum_{d ∣ n} f (d)$ for all $n$ if and only if $f (n) = \sum_{d ∣ n} g (d) μ (n / d)$ for all $n$ .

Proof. We first establish $μ * 1 = ε$ . At $n = 1$ the only divisor is $1$ , so $(μ * 1) (1) = μ (1) = 1 = ε (1)$ . Fix $n > 1$ and write its prime factorisation $n = p_{1}^{a_{1}} \dots p_{r}^{a_{r}}$ with $r \geq 1$ . Because $μ (d) = 0$ whenever $d$ has a repeated prime factor, the only divisors $d$ of $n$ contributing to $\sum_{d ∣ n} μ (d)$ are the squarefree ones, namely the products of subsets of ${p_{1}, \dots, p_{r}}$ . A subset of size $j$ yields a divisor $d$ with $μ (d) = (- 1)^{j}$ , and there are $(j r)$ such subsets. Hence $$ \sum_{d \mid n} \mu(d) = \sum_{j=0}^{r} \binom{r}{j} (-1)^j = (1 - 1)^r = 0, $$ by the binomial theorem, since $r \geq 1$ . This proves $μ * 1 = ε$ .

Now suppose $g = f * 1$ . Convolving on the right with $μ$ and using associativity and commutativity of $*$ together with $1 * μ = ε$ : $$ g * \mu = (f * \mathbf{1}) * \mu = f * (\mathbf{1} * \mu) = f * \varepsilon = f. $$ Conversely, if $f = g * μ$ , then $f * 1 = (g * μ) * 1 = g * (μ * 1) = g * ε = g$ . Writing the convolutions out as divisor sums gives the stated formulas, where $g * μ$ at $n$ is $\sum_{d ∣ n} g (d) μ (n / d)$ . $□$

Bridge. Möbius inversion builds toward 21.12.01 the prime number theorem, where the von Mangoldt identity $Λ = μ * lo g$ — itself an inversion of $lo g = Λ * 1$ — converts the prime-counting problem into the analytic study of $\sum Λ (n) n^{- s} = - ζ^{'} (s) / ζ (s)$ , and it appears again in 21.13.01 Dirichlet $L$ -functions, where the same inversion organises characters. The foundational reason the formula works is that $1$ is a unit of the convolution ring with inverse exactly $μ$ ; this is exactly the statement that summing over divisors and Möbius-weighting are inverse linear operations. The construction generalises: any arithmetic function $h$ with $h (1) \neq = 0$ has a Dirichlet inverse, so inversion against $1$ is one instance of a ring-theoretic fact. Putting these together, the totient identity $φ = μ * N$ (Master tier below) is read off as the Möbius inversion of $N = φ * 1$ , and the bridge from the multiplicative arithmetic of 21.01.02 to the Dirichlet-series analysis of 21.03.01 runs through this single ring.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Prove that the Dirichlet convolution is associative: $(f * g) * h = f * (g * h)$ for all $f, g, h \in A$ .

Hint

Both sides equal a sum over ordered triples $(a, b, c)$ with $ab c = n$ . Expand each side and match.

Answer

Evaluate $((f * g) * h) (n)$ . By definition it is $\sum_{d e = n} (f * g) (d) h (e) = \sum_{d e = n} (\sum_{ab = d} f (a) g (b)) h (e)$ . Replacing $d$ by $ab$ collapses the two layers into a single sum over ordered triples $(a, b, e)$ of positive integers with $ab e = n$ : $$ ((f * g) * h)(n) = \sum_{abe = n} f(a), g(b), h(e). $$ The same expansion of $(f * (g * h)) (n) = \sum_{a d = n} f (a) (g * h) (d) = \sum_{a d = n} f (a) \sum_{b c = d} g (b) h (c)$ gives $\sum_{ab c = n} f (a) g (b) h (c)$ , the identical triple sum (rename $c$ to $e$ ). Since both equal $\sum_{ab e = n} f (a) g (b) h (e)$ , they are equal. Rubric: full credit for reducing both sides to the symmetric triple sum; commutativity follows similarly from the symmetry of the pair sum $\sum_{ab = n} f (a) g (b)$ .

Exercise 4 (medium, symbolic).

Show that if $f \in A$ satisfies $f (1) \neq = 0$ , then $f$ has a Dirichlet inverse $f^{- 1}$ with $f * f^{- 1} = ε$ , given by the recursion $f^{- 1} (1) = 1/ f (1)$ and, for $n > 1$ , $$ f^{-1}(n) = -\frac{1}{f(1)} \sum_{\substack{d \mid n \ d < n}} f!\left(\frac{n}{d}\right) f^{-1}(d). $$

Hint

Impose $(f * f^{- 1}) (n) = ε (n)$ and solve for the top term $f^{- 1} (n)$ by isolating the $d = n$ summand.

Answer

Define $f^{- 1}$ by the stated recursion; it is well-defined because the value at $n$ depends only on values $f^{- 1} (d)$ with $d < n$ . We verify $(f * f^{- 1}) (n) = ε (n)$ by induction on $n$ . At $n = 1$ : $(f * f^{- 1}) (1) = f (1) f^{- 1} (1) = f (1) \cdot 1/ f (1) = 1 = ε (1)$ . For $n > 1$ , write the convolution isolating the $d = n$ term (which contributes $f (1) f^{- 1} (n)$ ): $$ (f * f^{-1})(n) = \sum_{d \mid n} f(n/d) f^{-1}(d) = f(1) f^{-1}(n) + \sum_{\substack{d \mid n \ d < n}} f(n/d) f^{-1}(d). $$ Substituting the recursion for $f^{- 1} (n)$ cancels the second sum exactly, leaving $0 = ε (n)$ . Hence $f * f^{- 1} = ε$ . Uniqueness follows because $ε$ is the identity and the convolution ring is commutative: any left inverse equals the right inverse. Rubric: full credit for the recursion verification plus the $f (1) \neq = 0$ well-definedness observation.

Exercise 5 (medium, symbolic).

Prove that $Λ = μ * lo g$ , where $Λ$ is the von Mangoldt function and $lo g (n) = lo g n$ .

Hint

First show $lo g = Λ * 1$ by computing $\sum_{d ∣ n} Λ (d)$ from the prime factorisation, then apply Möbius inversion.

Answer

Start from $(Λ * 1) (n) = \sum_{d ∣ n} Λ (d)$ . The nonzero terms come from prime-power divisors $p^{m} ∣ n$ , each contributing $lo g p$ . If $n = \prod_{i} p_{i}^{a_{i}}$ , then for each prime $p_{i}$ the divisors $p_{i}, p_{i}^{2}, \dots, p_{i}^{a_{i}}$ each contribute $lo g p_{i}$ , totalling $a_{i} lo g p_{i}$ . Summing over $i$ : $$ \sum_{d \mid n} \Lambda(d) = \sum_i a_i \log p_i = \log \prod_i p_i^{a_i} = \log n. $$ Thus $lo g = Λ * 1$ . By Möbius inversion (the Key Theorem, with $g = lo g$ , $f = Λ$ ), $Λ = lo g * μ = μ * lo g$ . Explicitly $Λ (n) = \sum_{d ∣ n} μ (n / d) lo g d = - \sum_{d ∣ n} μ (d) lo g d$ . Rubric: full credit for establishing $lo g = Λ * 1$ and inverting.

Exercise 7 (hard, symbolic).

Prove that the Dirichlet convolution of two multiplicative functions is multiplicative.

Hint

For coprime $m, n$ , every divisor of $mn$ factors uniquely as $d = d_{1} d_{2}$ with $d_{1} ∣ m$ , $d_{2} ∣ n$ , and $g cd (d_{1}, d_{2}) = 1$ . Split the convolution sum accordingly.

Answer

Let $f, g$ be multiplicative and $h = f * g$ . First $h (1) = f (1) g (1) = 1$ . Now fix coprime $m, n$ . Because $g cd (m, n) = 1$ , each divisor $d$ of $mn$ factors uniquely as $d = d_{1} d_{2}$ with $d_{1} ∣ m$ and $d_{2} ∣ n$ ; moreover $g cd (d_{1}, d_{2}) = 1$ and $g cd (m / d_{1}, n / d_{2}) = 1$ . Therefore $$ h(mn) = \sum_{d \mid mn} f(d) g(mn/d) = \sum_{d_1 \mid m} \sum_{d_2 \mid n} f(d_1 d_2), g!\left(\frac{m}{d_1} \cdot \frac{n}{d_2}\right). $$ Multiplicativity of $f$ on the coprime pair $(d_{1}, d_{2})$ and of $g$ on the coprime pair $(m / d_{1}, n / d_{2})$ gives $f (d_{1} d_{2}) = f (d_{1}) f (d_{2})$ and $g ((m / d_{1}) (n / d_{2})) = g (m / d_{1}) g (n / d_{2})$ . The double sum factors: $$ h(mn) = \Big( \sum_{d_1 \mid m} f(d_1) g(m/d_1) \Big) \Big( \sum_{d_2 \mid n} f(d_2) g(n/d_2) \Big) = h(m), h(n). $$ Hence $h$ is multiplicative. Rubric: full credit requires the unique coprime factorisation of divisors and the factoring of the double sum. Note the result fails for completely multiplicative inputs: $1 * 1 = d$ is multiplicative but not completely multiplicative, since $d (4) = 3 \neq = d (2)^{2} = 4$ .

Exercise 8 (hard, symbolic).

For a multiplicative function $f$ , prove the formal Dirichlet-series Euler product $$ \sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \prod_{p} \left( \sum_{m=0}^{\infty} \frac{f(p^m)}{p^{ms}} \right), $$ as an identity of formal Dirichlet series (equivalently, of absolutely convergent series where $\sum ∣ f (n) ∣ n^{- σ} < \infty$ ).

Hint

Expand the product over primes, picking one term $f (p^{m_{p}}) p^{- m_{p} s}$ from each factor; multiplicativity assembles these into $f (n) n^{- s}$ for $n = \prod p^{m_{p}}$ .

Answer

Work in the ring of formal Dirichlet series (or assume absolute convergence, which justifies rearrangement). Each factor is the Bell series $\sum_{m \geq 0} f (p^{m}) p^{- m s}$ . Forming the product over all primes and expanding, a general term selects, for each prime $p$ , a nonnegative exponent $m_{p}$ (all but finitely many zero), contributing $$ \prod_p f(p^{m_p}) p^{-m_p s}. $$ Set $n = \prod_{p} p^{m_{p}}$ . By unique factorisation each $n \geq 1$ arises from exactly one such selection. Multiplicativity of $f$ across the coprime prime powers gives $\prod_{p} f (p^{m_{p}}) = f (\prod_{p} p^{m_{p}}) = f (n)$ , and $\prod_{p} p^{- m_{p} s} = n^{- s}$ . Therefore the expanded product equals $\sum_{n \geq 1} f (n) n^{- s}$ , proving the Euler product. The convolution-to-multiplication dictionary makes this structural: if $D (f; s) = \sum_{n} f (n) n^{- s}$ denotes the Dirichlet series, then $D (f * g; s) = D (f; s) D (g; s)$ , and the Euler product is the statement that $D$ sends multiplicative functions to products over primes. Rubric: full credit for the unique-factorisation bijection between product terms and integers, plus the multiplicativity step; bonus for stating $D (f * g) = D (f) D (g)$ .

Lean formalization Intermediate+

Mathlib carries the convolution ring of arithmetic functions and the Möbius machinery, so the statements of this unit are expressible directly. The companion notes record the load-bearing declarations against Mathlib.NumberTheory.ArithmeticFunction.

-- Operative imports: Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheory.ArithmeticFunction
open ArithmeticFunction
open scoped ArithmeticFunction.Moebius ArithmeticFunction.zeta

-- Dirichlet convolution is the `*` of the CommRing instance on ArithmeticFunction R.
#check (inferInstance : CommRing (ArithmeticFunction ℂ))

-- The convolution identity ε is the `1` of the ring (one n = if n = 1 then 1 else 0).
example : (1 : ArithmeticFunction ℂ) 1 = 1 := by simp

-- μ * ζ = ε : the Möbius function is the Dirichlet inverse of the unit function 𝟙
-- (Mathlib's `zeta` is the unit function, value 1 on positive n, 0 on 0).
#check @ArithmeticFunction.moebius_mul_coe_zeta
-- ArithmeticFunction.moebius_mul_coe_zeta : μ * ↑ζ = 1

-- Möbius inversion, abstract form over an additive commutative group:
#check @ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq
-- (∀ n, 0 < n → g n = ∑ d ∈ n.divisors, f d) ↔
--   (∀ n, 0 < n → f n = ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2)

-- The von Mangoldt identity Λ * ζ = log, equivalently Λ = μ * log:
#check @ArithmeticFunction.vonMangoldt_mul_zeta
-- ArithmeticFunction.vonMangoldt_mul_zeta : Λ * ↑ζ = log

-- Multiplicativity is preserved by convolution:
#check @ArithmeticFunction.IsMultiplicative.mul
-- IsMultiplicative f → IsMultiplicative g → IsMultiplicative (f * g)

The gap documented in the unit metadata's Mathlib gap analysis is not the absence of these objects — Mathlib has them — but the absence of a single pedagogical module that proves the Dirichlet-inverse existence recursion, the subgroup structure of multiplicative functions under the units, the Bell-series criterion, and the formal-Dirichlet-series ring homomorphism with its Euler factorisation, end to end and in one place.

Advanced results Master

Theorem 1 (the convolution ring; Cashwell-Everett 1959). The ring $(A, +, *)$ of arithmetic functions over $C$ under pointwise addition and Dirichlet convolution is a commutative ring with identity $ε$ ; its group of units is ${f : f (1) \neq = 0}$ , with inverses given by the recursion of Exercise 4 ^{[Apostol Ch. 2]}. More is true: Cashwell and Everett 1959 Pacific J. Math. 9 proved that $A$ over a field is an integral domain, indeed a unique factorisation domain, isomorphic to the ring of formal power series in countably many variables $C [[x_{1}, x_{2}, \dots]]$ (one variable per prime), via $f \mapsto \sum f (n) \prod x_{i}^{a_{i}}$ for $n = \prod p_{i}^{a_{i}}$ . The absence of zero divisors follows from the order $ord (f) = min {n : f (n) \neq = 0}$ , which is multiplicative under convolution: $ord (f * g) = ord (f) \cdot ord (g)$ in the sense that the least nonzero index multiplies.

Theorem 2 (multiplicative functions form a group). The multiplicative functions, together with $ε$ excluded in favour of $1$ as the multiplicative-unit reference, form an abelian group under Dirichlet convolution: if $f, g$ are multiplicative then $f * g$ is multiplicative (Exercise 7), and the Dirichlet inverse $f^{- 1}$ of a multiplicative $f$ is again multiplicative. The completely multiplicative functions are not closed under convolution — $1 * 1 = d$ is multiplicative but not completely so — but a completely multiplicative $f$ has the especially simple inverse $f^{- 1} = μ f$ (pointwise product), since $(μ f) * f = ε$ by a direct check using $f (d) f (n / d) = f (n)$ . This recovers $1^{- 1} = μ$ as the case $f = 1$ .

Theorem 3 (the totient identities). The Euler totient satisfies $N = φ * 1$ , equivalently Gauss's identity $\sum_{d ∣ n} φ (d) = n$ ; Möbius inversion then yields $φ = N * μ$ , i.e. $φ (n) = \sum_{d ∣ n} μ (d) (n / d) = n \prod_{p ∣ n} (1 - 1/ p)$ . The product form follows because $φ = μ * N$ is a convolution of multiplicative functions, hence multiplicative, and on a prime power $φ (p^{a}) = p^{a} - p^{a - 1} = p^{a} (1 - 1/ p)$ . The Dirichlet series is $\sum_{n} φ (n) n^{- s} = ζ (s - 1) / ζ (s)$ for $Re (s) > 2$ , read off from $D (φ) = D (μ) D (N) = ζ (s)^{- 1} \cdot ζ (s - 1)$ via the dictionary $D (f * g) = D (f) D (g)$ and $D (μ) = 1/ ζ$ ^{[Hardy-Wright]}.

Theorem 4 (Dirichlet series dictionary and Euler products). The assignment $D : f \mapsto \sum_{n} f (n) n^{- s}$ is a ring homomorphism from $(A, +, *)$ to the ring of formal Dirichlet series: $D (f + g) = D (f) + D (g)$ and $D (f * g) = D (f) D (g)$ . Under $D$ , the distinguished functions map to recognisable series: $D (1) = ζ (s)$ , $D (ε) = 1$ , $D (μ) = 1/ ζ (s)$ , $D (N^{k}) = ζ (s - k)$ , $D (d) = ζ (s)^{2}$ , $D (σ_{k}) = ζ (s) ζ (s - k)$ , $D (λ) = ζ (2 s) / ζ (s)$ , and $D (Λ) = - ζ^{'} (s) / ζ (s)$ (from $Λ = μ * lo g$ and $D (lo g) = - ζ^{'} (s)$ ). For multiplicative $f$ , $D (f)$ factors as the Euler product $\prod_{p} (\sum_{m \geq 0} f (p^{m}) p^{- m s})$ (Exercise 8); for completely multiplicative $f$ the Bell series collapses to a geometric series, giving the single-factor Euler product $\prod_{p} (1 - f (p) p^{- s})^{- 1}$ .

Theorem 5 (Bell series and local multiplicativity). For a prime $p$ and a multiplicative $f$ , the Bell series is the formal power series $f_{p} (x) = \sum_{m \geq 0} f (p^{m}) x^{m}$ . Bell series convert Dirichlet convolution into ordinary power-series multiplication locally: $(f * g)_{p} (x) = f_{p} (x) g_{p} (x)$ for multiplicative $f, g$ ^[McCarthy]. This reduces identities among multiplicative functions to identities among power series, prime by prime. For instance $μ_{p} (x) = 1 - x$ , $1_{p} (x) = (1 - x)^{- 1}$ , and $(μ * 1)_{p} (x) = (1 - x) (1 - x)^{- 1} = 1 = ε_{p} (x)$ , recovering $μ * 1 = ε$ at the level of each prime; and $d_{p} (x) = (1 - x)^{- 2}$ confirms $d = 1 * 1$ .

Theorem 6 (averages and the hyperbola method). Convolution structure controls average orders. Dirichlet's hyperbola method 1849 ^{[Dirichlet 1849]} computes $\sum_{n \leq x} d (n) = x ln x + (2 γ - 1) x + O (x)$ by exploiting $d = 1 * 1$ and summing $\sum_{ab \leq x} 1$ over the hyperbola $ab \leq x$ , splitting at $a, b \leq x$ to avoid double counting. Similarly $\sum_{n \leq x} φ (n) = \frac{3}{π ^{2}} x^{2} + O (x ln x)$ , where the constant $3/ π^{2} = 1/ (2 ζ (2))$ comes from $φ = μ * N$ and the value $\sum μ (n) / n^{2} = 1/ ζ (2) = 6/ π^{2}$ . The Möbius function's own partial sums $M (x) = \sum_{n \leq x} μ (n)$ — the Mertens function — encode the prime number theorem: $M (x) = o (x)$ is equivalent to PNT, and $M (x) = O (x^{1/2 + ϵ})$ is equivalent to the Riemann hypothesis.

Synthesis. The Dirichlet convolution is the foundational reason the disparate arithmetic functions of 21.01.02 and 21.01.04 form a single coherent theory rather than a list of definitions, and this is exactly the bridge from elementary multiplicativity to the analytic machinery of 21.03.01. The central insight is that $(A, +, *)$ is a commutative ring in which $1$ is a unit with inverse $μ$ ; Möbius inversion is then nothing more than convolving with that inverse, and every classical inversion identity — $φ = μ * N$ , $Λ = μ * lo g$ , the Liouville-square identity — is one instance. Putting these together, the Dirichlet-series homomorphism $D (f * g) = D (f) D (g)$ generalises the ring structure into analysis: convolution becomes multiplication of series, multiplicativity becomes the Euler product, and the position of the zeros of $ζ$ becomes the position of the cancellation in $M (x) = \sum μ (n)$ . The pattern is dual to harmonic analysis on the multiplicative monoid $(Z_{> 0}, \times)$ : the Bell series localise convolution at each prime exactly as a Fourier transform diagonalises ordinary convolution, and the Euler product is the statement that the monoid is freely generated by the primes. This freeness, established by unique factorisation in 21.01.02, is what makes the whole apparatus run, and it generalises to the convolution algebras of $L$ -functions in 21.13.01 and the Hecke algebras of 21.04.01, where the same convolution-to-Euler-product dictionary organises automorphic objects.

Full proof set Master

Proposition 1 (completely multiplicative inverse). If $f$ is completely multiplicative, then its Dirichlet inverse is $f^{- 1} = μ f$ , the pointwise product.

Proof. Compute $(μ f * f) (n) = \sum_{d ∣ n} μ (d) f (d) f (n / d)$ . For completely multiplicative $f$ we have $f (d) f (n / d) = f (d \cdot n / d) = f (n)$ for every divisor $d$ , so the factor $f (n)$ pulls out: $$ (\mu f * f)(n) = f(n) \sum_{d \mid n} \mu(d) = f(n), \varepsilon(n). $$ For $n = 1$ this is $f (1) \cdot 1 = 1 = ε (1)$ ; for $n > 1$ it is $f (n) \cdot 0 = 0 = ε (n)$ . Hence $μ f * f = ε$ , so $f^{- 1} = μ f$ . The step $f (d) f (n / d) = f (n)$ uses complete multiplicativity without any coprimality hypothesis, which is why the simple formula holds only for completely multiplicative $f$ (for merely multiplicative $f$ the inverse is more involved). $□$

Proposition 2 (Dirichlet inverse of a multiplicative function is multiplicative). If $f$ is multiplicative with $f (1) = 1$ , its Dirichlet inverse $f^{- 1}$ is multiplicative.

Proof. Let $g$ be the multiplicative function defined on prime powers by the local inverse: on each prime power $p^{a}$ , set $g (p^{a})$ by the recursion of Exercise 4 applied to the Bell series, and extend multiplicatively to all $n$ via $g (\prod p^{a_{p}}) = \prod g (p^{a_{p}})$ . By Exercise 7 the convolution $f * g$ of two multiplicative functions is multiplicative, and on each prime power $(f * g) (p^{a}) = \sum_{j = 0}^{a} f (p^{j}) g (p^{a - j}) = ε (p^{a})$ by construction of $g$ . A multiplicative function is determined by its values on prime powers, and $ε$ is multiplicative with $ε (p^{a}) = 0$ for $a \geq 1$ , $ε (1) = 1$ ; since $f * g$ and $ε$ agree on all prime powers and both are multiplicative, $f * g = ε$ . Therefore $g = f^{- 1}$ and $f^{- 1}$ is multiplicative. $□$

Proposition 3 (Bell-series multiplicativity). For multiplicative $f, g$ and a prime $p$ , $(f * g)_{p} (x) = f_{p} (x) g_{p} (x)$ as formal power series.

Proof. The coefficient of $x^{m}$ in $f_{p} (x) g_{p} (x)$ is $\sum_{j = 0}^{m} f (p^{j}) g (p^{m - j})$ . Since the divisors of $p^{m}$ are exactly $1, p, \dots, p^{m}$ , the convolution at $p^{m}$ is $(f * g) (p^{m}) = \sum_{d ∣ p^{m}} f (d) g (p^{m} / d) = \sum_{j = 0}^{m} f (p^{j}) g (p^{m - j})$ , the same coefficient. Hence the power series $(f * g)_{p}$ and $f_{p} g_{p}$ have identical coefficients. $□$

Proposition 4 (Möbius inversion in the dual/multiplicative form). Let $F, G : Z_{> 0} \to C$ . Then $G (n) = \prod_{d ∣ n} F (d)$ for all $n$ if and only if $F (n) = \prod_{d ∣ n} G (d)^{μ (n / d)}$ for all $n$ .

Proof. Take logarithms: set $f = lo g \circ F$ and $g = lo g \circ G$ as arithmetic functions (assuming $F, G$ positive-real, or working in a fixed branch). The hypothesis $G = \prod_{d ∣ n} F (d)$ becomes $g = f * 1$ additively. By the Key Theorem, $g = f * 1 ⟺ f = g * μ$ , i.e. $lo g F (n) = \sum_{d ∣ n} μ (n / d) lo g G (d)$ . Exponentiating gives $F (n) = \prod_{d ∣ n} G (d)^{μ (n / d)}$ . The equivalence is exactly the additive Möbius inversion transported across the logarithm, which is a ring isomorphism from $(R_{> 0}, \times)$ to $(R, +)$ applied pointwise. $□$

Connections Master

The convolution ring is built on the unique factorisation of 21.01.02: the freeness of the multiplicative monoid $(Z_{> 0}, \times)$ on the primes is what makes Bell-series localisation and the Euler product valid, and the Möbius function is the inversion datum for the partial order of divisibility studied there.
The Dirichlet-series dictionary $D (f * g) = D (f) D (g)$ feeds directly into 21.03.01, where $D (1) = ζ (s)$ and $D (μ) = 1/ ζ (s)$ ; the analytic continuation and Euler product of $ζ$ developed there are the analytic shadow of the algebraic identities $1 * μ = ε$ and the multiplicativity proven here.
The von Mangoldt identity $Λ = μ * lo g$ established here is the entry point to 21.12.01 the prime number theorem, where $D (Λ) = - ζ^{'} / ζ$ converts the prime-counting average $ψ (x) = \sum_{n \leq x} Λ (n)$ into a contour integral controlled by the zeros of $ζ$ .
Characters and twisted convolutions generalise this ring in 21.13.01 Dirichlet $L$ -functions and characters, where multiplicative functions twisted by a Dirichlet character $χ$ retain the Euler-product structure and Möbius inversion organises the orthogonality relations.
The Euler totient identities $N = φ * 1$ and $φ = μ * N$ specialise the elementary congruence theory of 21.01.04 (Fermat-Euler), where $φ (n)$ is the order of the unit group $(Z / n Z)^{\times}$ that powers Euler's theorem.

Historical & philosophical context Master

The Möbius function entered mathematics in August Ferdinand Möbius's 1832 Crelle paper Über eine besondere Art von Umkehrung der Reihen ^{[Möbius 1832]}, where it arose as the coefficient sequence inverting a class of infinite series, well before the divisor-convolution framing existed. Möbius worked with the inversion of formal series $\sum a_{n} x^{n}$ and was not thinking of an algebra on arithmetic functions; the recognition that his function is the Dirichlet inverse of the constant function $1$ came later, through the systematic study of what we now call Dirichlet multiplication.

That algebraic reframing is due to several hands in the mid-nineteenth century. Dirichlet's 1849 study of mean values ^{[Dirichlet 1849]} used divisor sums and the hyperbola method without naming the convolution as a ring operation. Liouville's 1857 notes ^{[Liouville 1857]} introduced the completely multiplicative function $λ$ and exploited convolution identities freely. The clean statement that arithmetic functions form a commutative ring under $*$ , with $μ$ the inverse of $1$ , was consolidated in the analytic-number-theory texts of the twentieth century; Apostol's 1976 treatment ^{[Apostol Ch. 2]} is the canonical pedagogical source, and the ring-theoretic depth — that $A$ over a field is a unique factorisation domain isomorphic to a power-series ring — was settled by Cashwell and Everett in 1959.

The conceptual payoff is the dictionary between an algebraic operation on arithmetic functions and an analytic operation on Dirichlet series. Riemann's 1859 treatment of $ζ (s)$ as a function of a complex variable supplied the analytic side; the convolution ring supplies the algebraic side; and the homomorphism $D$ connecting them is the reason that statements about prime distribution can be translated into statements about a meromorphic function and back.

Bibliography Master

@book{apostol1976ant,
  author    = {Apostol, Tom M.},
  title     = {Introduction to Analytic Number Theory},
  series    = {Undergraduate Texts in Mathematics},
  publisher = {Springer-Verlag},
  year      = {1976},
  note      = {Chapter 2: arithmetical functions and Dirichlet multiplication}
}

@article{moebius1832umkehrung,
  author  = {M\"{o}bius, August Ferdinand},
  title   = {\"{U}ber eine besondere Art von Umkehrung der Reihen},
  journal = {Journal f\"{u}r die reine und angewandte Mathematik (Crelle)},
  volume  = {9},
  pages   = {105--123},
  year    = {1832}
}

@article{liouville1857fonctions,
  author  = {Liouville, Joseph},
  title   = {Sur quelques fonctions num\'{e}riques},
  journal = {Journal de Math\'{e}matiques Pures et Appliqu\'{e}es (2)},
  volume  = {2},
  pages   = {56--64},
  year    = {1857}
}

@article{cashwell1959ufd,
  author  = {Cashwell, E. D. and Everett, C. J.},
  title   = {The ring of number-theoretic functions},
  journal = {Pacific Journal of Mathematics},
  volume  = {9},
  number  = {4},
  pages   = {975--985},
  year    = {1959}
}

@book{hardywright2008,
  author    = {Hardy, G. H. and Wright, E. M.},
  title     = {An Introduction to the Theory of Numbers},
  edition   = {6},
  publisher = {Oxford University Press},
  year      = {2008},
  note      = {Revised by D. R. Heath-Brown and J. H. Silverman}
}

@book{mccarthy1986,
  author    = {McCarthy, Paul J.},
  title     = {Introduction to Arithmetical Functions},
  series    = {Universitext},
  publisher = {Springer-Verlag},
  year      = {1986}
}

Prerequisites

21.01.01
21.01.02
21.01.04

Tier anchors

beginner: Burton 2010 *Elementary Number Theory* 7e §6 (the divisor functions and the Möbius function, worked numerically); 3Blue1Brown 'Pi hiding in prime regularities' for the multiplicative-function viewpoint
intermediate: Apostol 1976 *Introduction to Analytic Number Theory* (Springer UTM) Ch. 2 (arithmetical functions, Dirichlet multiplication, Möbius inversion); Hardy-Wright 2008 *An Introduction to the Theory of Numbers* 6e (Oxford) Ch. XVI-XVII
master: Apostol 1976 *Introduction to Analytic Number Theory* Ch. 2-3 (the Dirichlet-convolution ring, Bell series, averages); Möbius 1832 *Crelle* 9 (originator of the inversion function); Dedekind 1857 and Liouville 1857 (early systematic use of the convolution and multiplicativity); McCarthy 1986 *Introduction to Arithmetical Functions* (Springer Universitext); Tóth 2014 survey *Multiplicative arithmetic functions of several variables*

References

Apostol, T. M. — Introduction to Analytic Number Theory · Springer Undergraduate Texts in Mathematics (1976). Chapter 2 develops arithmetical functions, the Dirichlet product (convolution), the ring structure with the identity $\varepsilon$ and the Dirichlet inverse, the Möbius function $\mu$ and inversion, the Euler totient $\varphi = \mu * N$, the von Mangoldt function $\Lambda = \mu * \log$, multiplicativity preserved under convolution, and the connection to Dirichlet series and Euler products. Chapter 3 treats averages of arithmetical functions and Bell series.
Möbius, A. F. — Über eine besondere Art von Umkehrung der Reihen · *Journal für die reine und angewandte Mathematik (Crelle)* 9 (1832), 105-123. Introduces the function $\mu(n)$ and the inversion formula now bearing his name, in the context of inverting series rather than the convolution ring (which is the later reformulation due to Dedekind, Liouville, and Cesàro).
Hardy, G. H. & Wright, E. M. — An Introduction to the Theory of Numbers · Oxford University Press, 6th edition (2008), revised by D. R. Heath-Brown and J. H. Silverman. Chapters XVI-XVII treat arithmetical functions, multiplicativity, the Möbius function and inversion, the divisor and totient functions, and generating Dirichlet series.
Dirichlet, P. G. L. — Über die Bestimmung der mittleren Werthe in der Zahlentheorie · *Abhandlungen der Königlichen Preussischen Akademie der Wissenschaften zu Berlin* (1849), 69-83. The Dirichlet hyperbola method for the average order of the divisor function, $\sum_{n \leq x} d(n) = x \ln x + (2\gamma - 1) x + O(\sqrt{x})$, the prototype convolution-average estimate.
McCarthy, P. J. — Introduction to Arithmetical Functions · Springer Universitext (1986). A systematic monograph on the convolution ring of arithmetical functions, Dirichlet inverses, multiplicative and completely multiplicative functions, Ramanujan sums, and generalised (Narkiewicz / regular) convolutions.
Liouville, J. — Sur quelques fonctions numériques · *Journal de Mathématiques Pures et Appliquées* (2) 2 (1857), 56-64 and following notes. Introduces the completely multiplicative function $\lambda(n) = (-1)^{\Omega(n)}$ now called the Liouville function, with the identity $\sum_{d \mid n} \lambda(d) = 1$ if $n$ is a perfect square and $0$ otherwise.

Estimated time

beginner: 18m
intermediate: 42m
master: 85m