Number Theory, Lecture 3

Number Theory, Lecture 3 Number Theory, Lecture 3 Jan Snellman Arithmetical functions, Dirichlet convolution, Multiplicative functions, Arithmetical functions Möbiusinversion Definition Some common arithmetical functions Dirichlet 1 Convolution Jan Snellman Matrix interpretation Order, Norms, 1 Infinite sums Matematiska Institutionen Multiplicative LinköpingsUniversitet function Definition Euler φ

M¨obius inversion Multiplicativity is preserved by multiplication Matrix veriﬁcation Divisor functions Euler φ again µ itself

Link¨oping,spring 2019

Lecture notes availabe at course homepage http://courses.mai.liu.se/GU/TATA54/ Number Summary Theory, Lecture 3

Jan Snellman

Arithmetical functions Definition Some common Definition arithmetical functions 1 Arithmetical functions Dirichlet Euler φ Convolution Definition Matrix 3 Möbiusinversion interpretation Some common arithmetical Order, Norms, Multiplicativity is preserved by Infinite sums functions Multiplicative multiplication function Dirichlet Convolution Definition Matrix verification Euler φ Matrix interpretation Divisor functions Möbius Order, Norms, Infinite sums inversion Euler φ again Multiplicativity is preserved by multiplication 2 Multiplicative function µ itself Matrix verification Divisor functions Euler φ again µ itself Number Summary Theory, Lecture 3

Jan Snellman

Multiplicative 20 function

Deﬁnition 15 Euler φ

M¨obius 10 inversion Multiplicativity is preserved by 5 multiplication Matrix veriﬁcation Divisor functions

Euler φ again 5 10 15 20 25 µ itself Number Arithmetical functions deﬁned by prime factorization Theory, Lecture 3

Jan Snellman

Arithmetical functions Definition a1 ar Some common n = p ··· p , qi distinct primes arithmetical 1 r functions Dirichlet Convolution Liouville function λ, Möbiusfunction µ: Matrix interpretation Order, Norms, Infinite sums ω(n) = r

Multiplicative function Ω(n) = a1 + ··· + ar Definition Ω(n) Euler φ λ(n) = (−1) Möbius inversion λ(n) ω(n) = Ω(n) Multiplicativity is preserved by µ(n) = multiplication 0 otherwise Matrix verification Divisor functions Euler φ again µ itself Number Arithmetical functions related to divisors Theory, Lecture 3

Jan Snellman

Arithmetical functions Definition d number of divisors, σ sum of divisors, and you know Euler φ. Some common arithmetical functions Dirichlet Convolution d(n) = 1 Matrix interpretation k|n Order, Norms, X Infinite sums σ(n) = k Multiplicative function k|n Definition X Euler φ φ(n) = 1 Möbius inversion 1≤kVon Mangoldt function Λ, prime-counting function π, Legendre symbol Arithmetical n , p-valuation v . functions p p Definition Some common arithmetical functions k Dirichlet log q n = q , q prime Convolution Matrix Λ(n) = interpretation 0 otherwise Order, Norms, Infinite sums

Multiplicative π(n) = 1 function 1≤k≤n Definition X Euler φ k prime Möbius 0 n ≡ 0 mod p inversion Multiplicativity is n 2 preserved by = +1 n 6≡ 0 mod p and exists a such that n ≡ a mod p multiplication p Matrix verification  2 Divisor functions −1 n 6≡ 0 mod p and exists no a such that n ≡ a mod p Euler φ again µ itself v (n) = k, pk |n, pk+1 6 |n p  Number Important arithmetical functions (not standard notation) Theory, Lecture 3

Jan Snellman

Arithmetical functions Definition Some common arithmetical functions 1 n = 1 Dirichlet e(n) = Convolution Matrix 0 n > 1 interpretation Order, Norms, Infinite sums 0(n) = 0 Multiplicative function 1(n) = 1 often denoted by ζ Definition Euler φ I(n) = n Möbius inversion 1 n = i Multiplicativity is preserved by ei (n) = multiplication 0 n 6= i Matrix verification Divisor functions Euler φ again µ itself Number Theory, Lecture 3 Jan Snellman Definition

Arithmetical functions Let f , g be arithmetical functions. Then their Dirichlet convolution is another a.f., Definition defined by Some common arithmetical functions Dirichlet Convolution (f ∗ g)(n) = f (a)g(b) = f (k)g(n/k) = f (n/`)g(`) (DC) Matrix interpretation 1≤a,b≤n 1≤k≤n 1≤`≤n Order, Norms, abX=n Xk|n X`|n Infinite sums

Multiplicative function Definition Euler φ Example Möbius inversion Multiplicativity is preserved by multiplication (f ∗ g)(10) = f (1)g(10) + f (2)g(5) + f (5)g(2) + f (10)g(1) Matrix verification Divisor functions Euler φ again µ itself Number The algebra of aritmetical functions Theory, Lecture 3

Jan Snellman

Arithmetical functions Definition • Some common f ∗ (g ∗ h) = (f ∗ g) ∗ h arithmetical functions • Dirichlet f ∗ g = g ∗ f Convolution Matrix • interpretation There is a unit for this multiplication, e(1) = 1 , e(n) = 0 for n > 1 Order, Norms, Infinite sums • Not all a.f. are invertible Multiplicative function • We can add: (f + g)(n) = f (n) + g(n) Definition Euler φ • We can scale: (cf )(n) = cf (n) Möbius inversion • 0(n) = 0 is a zero vector Multiplicativity is preserved by multiplication • A -vector space with multiplication; an algebra. Matrix verification C Divisor functions Euler φ again µ itself Number Matrix interpretation Theory, Lecture 3

Jan Snellman

Arithmetical functions Definition Some common arithmetical • Let n ∈ and D(n) = { 1 ≤ k ≤ n k|n } be its divisors functions P Dirichlet Convolution • We want to understand a.f. restricted to D(n), in particular their Matrix interpretation multiplication Order, Norms, Infinite sums • Multiplicative Given a.f. f , form matrix A with rows and columns indexed by elems in D(n), function and A = f (j/i) if i|j, 0 otherwise Definition ij Euler φ • Similarly for a.f. g and matrix B Möbius inversion • Multiplicativity is Then AB is the matrix for f ∗ g preserved by multiplication Matrix verification Divisor functions Euler φ again µ itself Number Example Theory, Lecture 3

Jan Snellman • n = 12, D(n) as follows • f = 1 Arithmetical functions 12 • A = ?? Deﬁnition Some common • arithmetical A ∗ A = ?? functions Dirichlet Convolution Matrix interpretation Order, Norms, Inﬁnite sums 4 6

Multiplicative function Deﬁnition Euler φ

M¨obius inversion Multiplicativity is 2 3 preserved by multiplication Matrix veriﬁcation Divisor functions Euler φ again µ itself

• 1 Number Summation Theory, Lecture 3

Jan Snellman

Arithmetical • functions F (n) = (1 ∗ f )(n) = k|n f (k) Deﬁnition Some common • The summation of f arithmetical P functions Dirichlet • Sometimes F is known and we want to recover f Convolution Matrix interpretation • Order, Norms, Inﬁnite sums

Multiplicative F (1) = f (1) function Definition F (2) = f (1) + f (2) Euler φ Möbius F (3) = f (1) + f (3) inversion Multiplicativity is preserved by F (4) = f (1) + f (2) + f (4) multiplication Matrix verification . Divisor functions . Euler φ again µ itself Number Inverses Theory, Lecture 3 Jan Snellman Theorem −1 Arithmetical f has inverse g = f iff f (1) 6= 0 functions Definition Some common Proof. arithmetical functions Want f ∗ g = e, so (f ∗ g)(m) = 1 if m = 1, 0 otherwise. Gives Dirichlet Convolution Matrix interpretation 1 = (f ∗ g)(1) = f (1)g(1) Order, Norms, Infinite sums 0 = (f ∗ g)(2) = f (1)g(2) + f (2)g(1) Multiplicative function 0 = (f ∗ g)(3) = f (1)g(3) + f (3)g(1) Definition 0 = (f ∗ g)(4) = f (1)g(4) + f (2)g(2) + f (4)g(1) Euler φ

M¨obius 0 = (f ∗ g)(5) = f (1)g(5) + f (5)g(1) inversion . Multiplicativity is . preserved by . multiplication Matrix veriﬁcation 0 = (f ∗ g)(n) = f (1)g(n) + f (k)g(n/k) Divisor functions Euler φ again Xk|n µ itself 1

so, by induction, we can solve for g(n). Number Normed algebra Theory, Lecture 3 Jan Snellman Deﬁnition

Arithmetical functions If f 6= 0, then the order of f is Definition Some common arithmetical ord min functions (f ) = { n f (n) 6= 0 } Dirichlet Convolution Matrix interpretation and the norm Order, Norms, −ord(f ) Infinite sums kf k = 2 Multiplicative function Definition Euler φ Lemma Möbius inversion • f = f (n)e , i.e., the partial sums of this sum converge to f Multiplicativity is n n preserved by multiplication • if f (1) = 0 then e + f is invertible, with inverse given by convergent geometric Matrix verification P Divisor functions series: Euler φ again e µ itself = e − f + f ∗ f − f ∗ f ∗ f + ··· e + f Number Theory, Lecture 3 Definition Jan Snellman

Arithmetical • f is totally multiplicative if f (nm) = f (n)f (m) functions Definition • f is multiplicative if f (nm) = f (n)f (m) whenever gcd(n, m) = 1 Some common arithmetical functions Dirichlet Convolution Matrix Theorem interpretation aj Order, Norms, Let n = p , prime factorization. Then Infinite sums j j Multiplicative • If f mult then f (n) = f (pj ), i.e., f is determined by its values at prime function Q j Definition powers Euler φ Q j Möbius • If f tot mult then f (n) = f (p) , i.e., f is determined by its values at primes inversion j Multiplicativity is preserved by multiplication Q Matrix verification Proof. Divisor functions Euler φ again µ itself Obvious! Number Theory, Lecture 3

Jan Snellman Theorem Arithmetical functions The Euler φ function is multiplicative. Deﬁnition Some common arithmetical functions Proof Dirichlet Convolution Matrix Let gcd(m, n) = 1. Want to prove φ(mn) = φ(m)φ(n), in other words, interpretation Order, Norms, Inﬁnite sums

Multiplicative |Zmn| = |Zm| |Zn| (1) function Definition Euler φ Claim: following bijection: Möbius inversion Multiplicativity is Zmn 3 [a]mn 7 ([a]m, [a]n) ∈ Zm × Zn (2) preserved by multiplication Matrix verification Divisor functions Euler φ again → µ itself Number Theory, Lecture 3 Jan Snellman Proof.

Arithmetical functions • Well-defined, since a ≡ a 0 mod mn implies a ≡ a 0 mod m and a ≡ a 0 Definition Some common mod arithmetical n. functions Dirichlet 0 mod 0 mod 0 mod Convolution • Injective, since a ≡ a m and a ≡ a n implies a ≡ a mn Matrix interpretation • Surjective, by the CRT: take c, d, then exists x with Order, Norms, Infinite sums Multiplicative mod function x ≡ c m Definition mod Euler φ x ≡ d n

Möbius inversion so [x] 7 ([c] , [d] ) Multiplicativity is mn m n preserved by multiplication Matrix verification Divisor functions → Euler φ again µ itself Number Theory, 1 Take p prime Lecture 3 2 Jan Snellman Then all 1 ≤ a < p relatively prime to p, so φ(p) = p − 1 3 Now consider prime power pr Arithmetical functions r gcd r Definition 4 For 1 ≤ a < p , (a, p ) > 1 iff p|n Some common arithmetical functions Dirichlet Convolution 2 5 8 Matrix interpretation 1 4 7 Order, Norms, 3 6 9 Infinite sums 5 Example: p = 3, r = 2: Multiplicative r r pr r 1 function 6 So φ(p ) = p − p = p 1 − p Definition Euler φ r1 rs 7 For n = p1 ··· ps , we have by multiplicativity Möbius inversion r1 rs r1 rs Multiplicativity is φ(p ··· p ) = φ(p ) ··· φ(p ) preserved by 1 s 1 s multiplication r1 rs Matrix verification = p1 ··· ps (1 − 1/p1) ··· (1 − 1/ps ) Divisor functions Euler φ again µ itself = n (1 − 1/pj ) j Y Number Theory, Lecture 3

Jan Snellman

Arithmetical functions Definition Some common Example arithmetical functions Dirichlet Convolution • Matrix φ(15) = φ(3)φ(5) = 2 ∗ 4 = 8 interpretation 4 4 3 Order, Norms, • Infinite sums φ(16) = φ(2 ) = 2 − 2 = 8 Multiplicative • φ(120) = φ(23 ∗ 3 ∗ 5) = 120(1 − 1/2)(1 − 1/3)(1 − 1/5) = 120 ∗ (4/15) = 32. function Definition Euler φ

M¨obius inversion Multiplicativity is preserved by multiplication Matrix veriﬁcation Divisor functions Euler φ again µ itself Number n = p gives φ(n) = n − 1. This is visible in graph of φ(n). Theory, Lecture 3

Jan Snellman 3000

Arithmetical functions Definition 2500 Some common arithmetical functions Dirichlet Convolution 2000 Matrix interpretation Order, Norms, Infinite sums 1500 Multiplicative function Definition Euler φ 1000 Möbius inversion Multiplicativity is preserved by multiplication 500 Matrix verification Divisor functions Euler φ again µ itself 500 1000 1500 2000 2500 3000 Number Theory, Theorem Lecture 3

Jan Snellman f , g (non-zero) multiplicative arithmetical functions, h = f ∗ g

Arithmetical functions (i) e is multiplicative (iii) h is multiplicative Definition −1 Some common (ii) f (1) = 1, so f is invertible (iv) f is multiplicative arithmetical functions Dirichlet Convolution Matrix interpretation Proof Order, Norms, Infinite sums (i-ii) Trivial. (iii): Suppose gcd(m, n) = 1. Then Multiplicative function Definition mn m n Euler φ h(mn) = (f ∗ g)(mn) = f (k)g( ) = f (k1k2)g( ) Möbius k k1 k2 inversion kX|mn kX1|m Multiplicativity is k2|n preserved by multiplication m n m n Matrix verification = f (k1)f (k2)g( )g( ) = f (k1)g( ) f (k2)g( ) = h(m)h(n) Divisor functions k1 k2 k1 k2 Euler φ again k1|m k1|m k2|n µ itself X X X k2|n Number Theory, Lecture 3

Jan Snellman Proof.

Arithmetical (iv): The formula for the inverse now becomes functions Deﬁnition nm Some common −1 −1 arithmetical f (n) = − f (d)f ( ) functions d Dirichlet d|n Convolution Xd

Arithmetical functions 1 1 ∗ µ = e Definition Some common 2 F (n) = f (k) for all n iff f (n) = F (k)µ(n/k) for all n arithmetical k|n k|n functions Dirichlet Convolution P P Matrix interpretation Proof. Order, Norms, Infinite sums (1): Since the a.f. involved are multiplicative (check!), it suffices to check on prime Multiplicative function powers pr . Then (1 ∗ µ)(p0) = 1, and for r > 0 Definition Euler φ r Möbius r k inversion (µ ∗ 1)(p ) = µ(p ) = 1 − 1 + 0 + ··· + 0 = 0. Multiplicativity is preserved by k=0 multiplication X Matrix verification Divisor functions (2): If F = f ∗ 1 then f = f ∗ e = f ∗ 1 ∗ µ = F ∗ µ. Euler φ again µ itself Number Theory, Example Lecture 3

Jan Snellman • n = 12, D(n) as follows • f = 1

Arithmetical 12 functions • A = ?? Deﬁnition • Some common g = µ arithmetical 4 6 functions • C = ?? Dirichlet Convolution • Matrix 2 3 AC = ?? interpretation Order, Norms, Inﬁnite sums

1 Multiplicative • function Deﬁnition Euler φ

Möbius inversion Multiplicativity is preserved by multiplication Matrix verification Divisor functions Euler φ again µ itself Number Theory, Lecture 3 Recall Jan Snellman d(n) = 1, σ(n) = k Arithmetical functions Xk|n Xk|n Definition Some common We can write this as arithmetical functions Dirichlet d = 1 ∗ 1, σ = 1 ∗ I Convolution Matrix interpretation from which we conclude that d, σ are multiplicative, and that Order, Norms, Infinite sums

Multiplicative function µ ∗ d = 1, µ ∗ σ = I Definition Euler φ or in other words Möbius inversion Multiplicativity is preserved by µ(k)d(n/k) = 1, µ(k)σ(n/k) = n multiplication Matrix verification k|n k|n Divisor functions X X Euler φ again µ itself Number Theory, Lecture 3 Definition Jan Snellman k σk (n) = d . In particular, σ0 = d, σ1 = σ. Arithmetical d|n functions Definition P Some common arithmetical Lemma functions Dirichlet Convolution σk is multiplicative Matrix interpretation Order, Norms, Infinite sums Proof. Multiplicative function Suppose gcd(m, n) = 1. Then Definition Euler φ k k k k k k Möbius σk (mn) = d = (d1d2) = d1 d2 = d1 d2 = σk (m)σk (n) inversion Multiplicativity is d|mn d1|m d1|m d1|m d2|n preserved by X X X X X multiplication d2|n d2|n Matrix verification Divisor functions Euler φ again µ itself Number Theory, Lecture 3

Jan Snellman

Arithmetical functions Theorem Definition Some common k(aj +1) arithmetical a r 1−p functions 1 σ (p 1 ··· par ) = j Dirichlet k 1 r j=1 1−pk Convolution j Matrix k k interpretation 2 d µ(n/d) =Qn Order, Norms, d|n Infinite sums Multiplicative P function Definition Proof. Euler φ

Möbius Try to prove it yourself! inversion Multiplicativity is preserved by multiplication Matrix verification Divisor functions Euler φ again µ itself Number Theory, Lecture 3 Jan Snellman Lemma Arithmetical functions 1 ∗ φ = I Definition Some common arithmetical functions Proof. Dirichlet Convolution Matrix In other words, want prove interpretation Order, Norms, Infinite sums φ(k) = n.

Multiplicative k|n function X r Definition Multiplicative, so put n = p . Euler φ LHS Möbius If r = 0: = 1, OK. inversion If r > 0: LHS = r φ(pj ) = 1 + r (pj − pj−1) = pr , since sum Multiplicativity is j=0 j=1 preserved by multiplication telescoping. Matrix verification P P Divisor functions Euler φ again µ itself Number Divisors of 12 Theory, Lecture 3

Jan Snellman

Arithmetical φ(1) + φ(2) + φ(3) + φ(6) + φ(12) = 1 + 1 + 2 + 2 + 2 + 4 = 12 functions Deﬁnition Some common arithmetical functions Dirichlet Convolution Matrix interpretation Order, Norms, Inﬁnite sums

Multiplicative function Deﬁnition Euler φ

M¨obius inversion Multiplicativity is preserved by multiplication Matrix veriﬁcation Divisor functions Euler φ again µ itself Number Theory, Lecture 3 Jan Snellman Theorem

Arithmetical functions n n Definition φ(n) = µ(k) = kµ( ) Some common k k arithmetical k|n k|n functions X X Dirichlet Convolution Matrix interpretation Order, Norms, Proof. Infinite sums Multiplicative Since function Definition 1 ∗ φ = I, Euler φ Möbius we have that inversion Multiplicativity is φ = µ ∗ I = I ∗ µ preserved by multiplication Matrix verification Divisor functions Euler φ again µ itself Number Theory, Lecture 3

Jan Snellman Definition n Arithmetical An n’th root of unity is a complex root to z = 1. A primitive n’th root of unity is functions Definition not a k’th root of unity for smaller k. Some common arithmetical functions Dirichlet Lemma Convolution Matrix exp 2π s interpretation Put ξn = ( n i). Then the n’th roots of unity are ξn, 1 ≤ s ≤ n, and the Order, Norms, Infinite sums k gcd primitive n’th roots of unity are ξn, (k, n) = 1. Multiplicative function Definition Lemma Euler φ Möbius If n > 1, inversion n n Multiplicativity is ξ − 1 preserved by s n multiplication ξn = = 0. ξn − 1 Matrix verification s=1 Divisor functions X Euler φ again µ itself Number Theory, Lecture 3 Lemma Jan Snellman n Arithmetical functions s ` 0 = ξn = ξn Definition Some common s=1 k|n gcd(`,k)=1 arithmetical functions X X X Dirichlet Convolution Matrix interpretation Order, Norms, Let f (d) denote the sum of the primitive Infinite sums

Multiplicative d’th roots of unity. Then f (1) = 1, and function for n > 1, f (d) = 0. So 1 ∗ f = e, Definition d|n Euler φ hence f = µ. So the Möbiusfunction is Möbius P inversion the sum of the primitive roots. Multiplicativity is preserved by multiplication Matrix verification Divisor functions Euler φ again µ itself