Documentation

Mathlib.NumberTheory.ArithmeticFunction

Arithmetic Functions and Dirichlet Convolution #

This file defines arithmetic functions, which are functions from to a specified type that map 0 to 0. In the literature, they are often instead defined as functions from ℕ+. These arithmetic functions are endowed with a multiplication, given by Dirichlet convolution, and pointwise addition, to form the Dirichlet ring.

Main Definitions #

Main Results #

Notation #

All notation is localized in the namespace ArithmeticFunction.

The arithmetic functions ζ, σ, ω, Ω and μ have Greek letter names.

In addition, there are separate locales ArithmeticFunction.zeta for ζ, ArithmeticFunction.sigma for σ, ArithmeticFunction.omega for ω, ArithmeticFunction.Omega for Ω, and ArithmeticFunction.Moebius for μ, to allow for selective access to these notations.

The arithmetic function $$n \mapsto \prod_{p \mid n} f(p)$$ is given custom notation ∏ᵖ p ∣ n, f p when applied to n.

Tags #

arithmetic functions, dirichlet convolution, divisors

def ArithmeticFunction (R : Type u_1) [Zero R] :
Type u_1

An arithmetic function is a function from that maps 0 to 0. In the literature, they are often instead defined as functions from ℕ+. Multiplication on ArithmeticFunctions is by Dirichlet convolution.

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    @[simp]
    theorem ArithmeticFunction.toFun_eq {R : Type u_1} [Zero R] (f : ArithmeticFunction R) :
    f.toFun = f
    @[simp]
    theorem ArithmeticFunction.coe_mk {R : Type u_1} [Zero R] (f : R) (hf : f 0 = 0) :
    { toFun := f, map_zero' := hf } = f
    @[simp]
    theorem ArithmeticFunction.map_zero {R : Type u_1} [Zero R] {f : ArithmeticFunction R} :
    f 0 = 0
    theorem ArithmeticFunction.coe_inj {R : Type u_1} [Zero R] {f : ArithmeticFunction R} {g : ArithmeticFunction R} :
    f = g f = g
    @[simp]
    theorem ArithmeticFunction.zero_apply {R : Type u_1} [Zero R] {x : } :
    0 x = 0
    theorem ArithmeticFunction.ext {R : Type u_1} [Zero R] ⦃f : ArithmeticFunction R ⦃g : ArithmeticFunction R (h : ∀ (x : ), f x = g x) :
    f = g
    theorem ArithmeticFunction.ext_iff {R : Type u_1} [Zero R] {f : ArithmeticFunction R} {g : ArithmeticFunction R} :
    f = g ∀ (x : ), f x = g x
    instance ArithmeticFunction.one {R : Type u_1} [Zero R] [One R] :
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    theorem ArithmeticFunction.one_apply {R : Type u_1} [Zero R] [One R] {x : } :
    1 x = if x = 1 then 1 else 0
    @[simp]
    theorem ArithmeticFunction.one_one {R : Type u_1} [Zero R] [One R] :
    1 1 = 1
    @[simp]
    theorem ArithmeticFunction.one_apply_ne {R : Type u_1} [Zero R] [One R] {x : } (h : x 1) :
    1 x = 0

    Coerce an arithmetic function with values in to one with values in R. We cannot inline this in natCoe because it gets unfolded too much.

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    • f = { toFun := fun (n : ) => (f n), map_zero' := (_ : (f 0) = 0) }
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      • ArithmeticFunction.natCoe = { coe := ArithmeticFunction.natToArithmeticFunction }
      @[simp]
      theorem ArithmeticFunction.natCoe_apply {R : Type u_1} [AddMonoidWithOne R] {f : ArithmeticFunction } {x : } :
      f x = (f x)

      Coerce an arithmetic function with values in to one with values in R. We cannot inline this in intCoe because it gets unfolded too much.

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      • f = { toFun := fun (n : ) => (f n), map_zero' := (_ : (f 0) = 0) }
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        • ArithmeticFunction.intCoe = { coe := ArithmeticFunction.ofInt }
        @[simp]
        theorem ArithmeticFunction.intCoe_apply {R : Type u_1} [AddGroupWithOne R] {f : ArithmeticFunction } {x : } :
        f x = (f x)
        @[simp]
        theorem ArithmeticFunction.coe_coe {R : Type u_1} [AddGroupWithOne R] {f : ArithmeticFunction } :
        f = f
        @[simp]
        @[simp]
        theorem ArithmeticFunction.intCoe_one {R : Type u_1} [AddGroupWithOne R] :
        1 = 1
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        • ArithmeticFunction.add = { add := fun (f g : ArithmeticFunction R) => { toFun := fun (n : ) => f n + g n, map_zero' := (_ : f 0 + g 0 = 0) } }
        @[simp]
        theorem ArithmeticFunction.add_apply {R : Type u_1} [AddMonoid R] {f : ArithmeticFunction R} {g : ArithmeticFunction R} {n : } :
        (f + g) n = f n + g n
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        • ArithmeticFunction.instAddMonoidWithOne = let src := ArithmeticFunction.instAddMonoid; let src_1 := ArithmeticFunction.one; AddMonoidWithOne.mk
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        The Dirichlet convolution of two arithmetic functions f and g is another arithmetic function such that (f * g) n is the sum of f x * g y over all (x,y) such that x * y = n.

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        @[simp]
        theorem ArithmeticFunction.smul_apply {R : Type u_1} {M : Type u_2} [Zero R] [AddCommMonoid M] [SMul R M] {f : ArithmeticFunction R} {g : ArithmeticFunction M} {n : } :
        (f g) n = Finset.sum (Nat.divisorsAntidiagonal n) fun (x : × ) => f x.1 g x.2

        The Dirichlet convolution of two arithmetic functions f and g is another arithmetic function such that (f * g) n is the sum of f x * g y over all (x,y) such that x * y = n.

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        • ArithmeticFunction.instMulArithmeticFunctionToZeroToMonoidWithZero = { mul := fun (x x_1 : ArithmeticFunction R) => x x_1 }
        @[simp]
        theorem ArithmeticFunction.mul_apply {R : Type u_1} [Semiring R] {f : ArithmeticFunction R} {g : ArithmeticFunction R} {n : } :
        (f * g) n = Finset.sum (Nat.divisorsAntidiagonal n) fun (x : × ) => f x.1 * g x.2
        theorem ArithmeticFunction.mul_apply_one {R : Type u_1} [Semiring R] {f : ArithmeticFunction R} {g : ArithmeticFunction R} :
        (f * g) 1 = f 1 * g 1
        @[simp]
        theorem ArithmeticFunction.natCoe_mul {R : Type u_1} [Semiring R] {f : ArithmeticFunction } {g : ArithmeticFunction } :
        (f * g) = f * g
        @[simp]
        theorem ArithmeticFunction.intCoe_mul {R : Type u_1} [Ring R] {f : ArithmeticFunction } {g : ArithmeticFunction } :
        (f * g) = f * g
        theorem ArithmeticFunction.mul_smul' {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (f : ArithmeticFunction R) (g : ArithmeticFunction R) (h : ArithmeticFunction M) :
        (f * g) h = f g h
        theorem ArithmeticFunction.one_smul' {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (b : ArithmeticFunction M) :
        1 b = b
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        • ArithmeticFunction.instCommSemiringArithmeticFunctionToZeroToCommMonoidWithZero = let src := ArithmeticFunction.instSemiring; CommSemiring.mk (_ : ∀ (f g : ArithmeticFunction R), f * g = g * f)
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        • One or more equations did not get rendered due to their size.

        ζ 0 = 0, otherwise ζ x = 1. The Dirichlet Series is the Riemann ζ.

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          ζ 0 = 0, otherwise ζ x = 1. The Dirichlet Series is the Riemann ζ.

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            ζ 0 = 0, otherwise ζ x = 1. The Dirichlet Series is the Riemann ζ.

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              @[simp]
              theorem ArithmeticFunction.zeta_apply {x : } :
              ArithmeticFunction.zeta x = if x = 0 then 0 else 1

              This is the pointwise product of ArithmeticFunctions.

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                @[simp]

                This is the pointwise power of ArithmeticFunctions.

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                  @[simp]
                  theorem ArithmeticFunction.ppow_apply {R : Type u_1} [Semiring R] {f : ArithmeticFunction R} {k : } {x : } (kpos : 0 < k) :

                  This is the pointwise division of ArithmeticFunctions.

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                    @[simp]

                    This result only holds for DivisionSemirings instead of GroupWithZeros because zeta takes values in ℕ, and hence the coercion requires an AddMonoidWithOne. TODO: Generalise zeta

                    The map $n \mapsto \prod_{p \mid n} f(p)$ as an arithmetic function

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                      ∏ᵖ p ∣ n, f p is custom notation for prodPrimeFactors f n

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                        @[simp]
                        theorem ArithmeticFunction.prodPrimeFactors_apply {R : Type u_1} [CommMonoidWithZero R] {f : R} {n : } (hn : n 0) :
                        (ArithmeticFunction.prodPrimeFactors fun (p : ) => f p) n = Finset.prod n.primeFactors fun (p : ) => f p

                        Multiplicative functions

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                          theorem ArithmeticFunction.IsMultiplicative.map_prod {R : Type u_1} {ι : Type u_2} [CommMonoidWithZero R] (g : ι) {f : ArithmeticFunction R} (hf : ArithmeticFunction.IsMultiplicative f) (s : Finset ι) (hs : Set.Pairwise (s) (Nat.Coprime on g)) :
                          f (Finset.prod s fun (i : ι) => g i) = Finset.prod s fun (i : ι) => f (g i)
                          theorem ArithmeticFunction.IsMultiplicative.map_prod_of_prime {R : Type u_1} [CommSemiring R] {f : ArithmeticFunction R} (h_mult : ArithmeticFunction.IsMultiplicative f) (t : Finset ) (ht : pt, Nat.Prime p) :
                          f (Finset.prod t fun (a : ) => a) = Finset.prod t fun (a : ) => f a
                          theorem ArithmeticFunction.IsMultiplicative.map_prod_of_subset_primeFactors {R : Type u_1} [CommSemiring R] {f : ArithmeticFunction R} (h_mult : ArithmeticFunction.IsMultiplicative f) (l : ) (t : Finset ) (ht : t l.primeFactors) :
                          f (Finset.prod t fun (a : ) => a) = Finset.prod t fun (a : ) => f a

                          For any multiplicative function f and any n > 0, we can evaluate f n by evaluating f at p ^ k over the factorization of n

                          theorem ArithmeticFunction.IsMultiplicative.iff_ne_zero {R : Type u_1} [MonoidWithZero R] {f : ArithmeticFunction R} :
                          ArithmeticFunction.IsMultiplicative f f 1 = 1 ∀ {m n : }, m 0n 0Nat.Coprime m nf (m * n) = f m * f n

                          A recapitulation of the definition of multiplicative that is simpler for proofs

                          Two multiplicative functions f and g are equal if and only if they agree on prime powers

                          The identity on as an ArithmeticFunction.

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                            @[simp]
                            theorem ArithmeticFunction.pow_apply {k : } {n : } :
                            (ArithmeticFunction.pow k) n = if k = 0 n = 0 then 0 else n ^ k

                            σ k n is the sum of the kth powers of the divisors of n

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                              σ k n is the sum of the kth powers of the divisors of n

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                                σ k n is the sum of the kth powers of the divisors of n

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                                  Ω n is the number of prime factors of n.

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                                    Ω n is the number of prime factors of n.

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                                      Ω n is the number of prime factors of n.

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                                        ω n is the number of distinct prime factors of n.

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                                          ω n is the number of distinct prime factors of n.

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                                            ω n is the number of distinct prime factors of n.

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                                              μ is the Möbius function. If n is squarefree with an even number of distinct prime factors, μ n = 1. If n is squarefree with an odd number of distinct prime factors, μ n = -1. If n is not squarefree, μ n = 0.

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                                                μ is the Möbius function. If n is squarefree with an even number of distinct prime factors, μ n = 1. If n is squarefree with an odd number of distinct prime factors, μ n = -1. If n is not squarefree, μ n = 0.

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                                                  μ is the Möbius function. If n is squarefree with an even number of distinct prime factors, μ n = 1. If n is squarefree with an odd number of distinct prime factors, μ n = -1. If n is not squarefree, μ n = 0.

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                                                    theorem ArithmeticFunction.moebius_apply_prime_pow {p : } {k : } (hp : Nat.Prime p) (hk : k 0) :
                                                    ArithmeticFunction.moebius (p ^ k) = if k = 1 then -1 else 0

                                                    A unit in ArithmeticFunction R that evaluates to ζ, with inverse μ.

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                                                      @[simp]
                                                      theorem ArithmeticFunction.coe_zetaUnit {R : Type u_1} [CommRing R] :
                                                      ArithmeticFunction.zetaUnit = ArithmeticFunction.zeta
                                                      @[simp]
                                                      theorem ArithmeticFunction.inv_zetaUnit {R : Type u_1} [CommRing R] :
                                                      ArithmeticFunction.zetaUnit⁻¹ = ArithmeticFunction.moebius
                                                      theorem ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq {R : Type u_1} [AddCommGroup R] {f : R} {g : R} :
                                                      (n > 0, (Finset.sum (Nat.divisors n) fun (i : ) => f i) = g n) n > 0, (Finset.sum (Nat.divisorsAntidiagonal n) fun (x : × ) => ArithmeticFunction.moebius x.1 g x.2) = f n

                                                      Möbius inversion for functions to an AddCommGroup.

                                                      theorem ArithmeticFunction.sum_eq_iff_sum_mul_moebius_eq {R : Type u_1} [Ring R] {f : R} {g : R} :
                                                      (n > 0, (Finset.sum (Nat.divisors n) fun (i : ) => f i) = g n) n > 0, (Finset.sum (Nat.divisorsAntidiagonal n) fun (x : × ) => (ArithmeticFunction.moebius x.1) * g x.2) = f n

                                                      Möbius inversion for functions to a Ring.

                                                      theorem ArithmeticFunction.prod_eq_iff_prod_pow_moebius_eq {R : Type u_1} [CommGroup R] {f : R} {g : R} :
                                                      (n > 0, (Finset.prod (Nat.divisors n) fun (i : ) => f i) = g n) n > 0, (Finset.prod (Nat.divisorsAntidiagonal n) fun (x : × ) => g x.2 ^ ArithmeticFunction.moebius x.1) = f n

                                                      Möbius inversion for functions to a CommGroup.

                                                      theorem ArithmeticFunction.prod_eq_iff_prod_pow_moebius_eq_of_nonzero {R : Type u_1} [CommGroupWithZero R] {f : R} {g : R} (hf : ∀ (n : ), 0 < nf n 0) (hg : ∀ (n : ), 0 < ng n 0) :
                                                      (n > 0, (Finset.prod (Nat.divisors n) fun (i : ) => f i) = g n) n > 0, (Finset.prod (Nat.divisorsAntidiagonal n) fun (x : × ) => g x.2 ^ ArithmeticFunction.moebius x.1) = f n

                                                      Möbius inversion for functions to a CommGroupWithZero.

                                                      theorem ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq_on {R : Type u_1} [AddCommGroup R] {f : R} {g : R} (s : Set ) (hs : ∀ (m n : ), m nn sm s) :
                                                      (n > 0, n s(Finset.sum (Nat.divisors n) fun (i : ) => f i) = g n) n > 0, n s(Finset.sum (Nat.divisorsAntidiagonal n) fun (x : × ) => ArithmeticFunction.moebius x.1 g x.2) = f n

                                                      Möbius inversion for functions to an AddCommGroup, where the equalities only hold on a well-behaved set.

                                                      theorem ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq_on' {R : Type u_1} [AddCommGroup R] {f : R} {g : R} (s : Set ) (hs : ∀ (m n : ), m nn sm s) (hs₀ : 0s) :
                                                      (ns, (Finset.sum (Nat.divisors n) fun (i : ) => f i) = g n) ns, (Finset.sum (Nat.divisorsAntidiagonal n) fun (x : × ) => ArithmeticFunction.moebius x.1 g x.2) = f n
                                                      theorem ArithmeticFunction.sum_eq_iff_sum_mul_moebius_eq_on {R : Type u_1} [Ring R] {f : R} {g : R} (s : Set ) (hs : ∀ (m n : ), m nn sm s) :
                                                      (n > 0, n s(Finset.sum (Nat.divisors n) fun (i : ) => f i) = g n) n > 0, n s(Finset.sum (Nat.divisorsAntidiagonal n) fun (x : × ) => (ArithmeticFunction.moebius x.1) * g x.2) = f n

                                                      Möbius inversion for functions to a Ring, where the equalities only hold on a well-behaved set.

                                                      theorem ArithmeticFunction.prod_eq_iff_prod_pow_moebius_eq_on {R : Type u_1} [CommGroup R] {f : R} {g : R} (s : Set ) (hs : ∀ (m n : ), m nn sm s) :
                                                      (n > 0, n s(Finset.prod (Nat.divisors n) fun (i : ) => f i) = g n) n > 0, n s(Finset.prod (Nat.divisorsAntidiagonal n) fun (x : × ) => g x.2 ^ ArithmeticFunction.moebius x.1) = f n

                                                      Möbius inversion for functions to a CommGroup, where the equalities only hold on a well-behaved set.

                                                      theorem ArithmeticFunction.prod_eq_iff_prod_pow_moebius_eq_on_of_nonzero {R : Type u_1} [CommGroupWithZero R] (s : Set ) (hs : ∀ (m n : ), m nn sm s) {f : R} {g : R} (hf : n > 0, f n 0) (hg : n > 0, g n 0) :
                                                      (n > 0, n s(Finset.prod (Nat.divisors n) fun (i : ) => f i) = g n) n > 0, n s(Finset.prod (Nat.divisorsAntidiagonal n) fun (x : × ) => g x.2 ^ ArithmeticFunction.moebius x.1) = f n

                                                      Möbius inversion for functions to a CommGroupWithZero, where the equalities only hold on a well-behaved set.