Number Theory, Lecture 3 Number Theory, Lecture 3 Jan Snellman Arithmetical functions, Dirichlet convolution, Multiplicative functions, Arithmetical functions M¨obiusinversion Definition Some common arithmetical functions Dirichlet 1 Convolution Jan Snellman Matrix interpretation Order, Norms, 1 Infinite sums Matematiska Institutionen Multiplicative Link¨opingsUniversitet 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 M¨obiusinversion 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 M¨obius 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 Arithmetical functions Definition Some common Definition arithmetical functions 1 Arithmetical functions Dirichlet Euler φ Convolution Definition Matrix 3 M¨obiusinversion 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 M¨obius 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 Arithmetical functions Definition Some common Definition arithmetical functions 1 Arithmetical functions Dirichlet Euler φ Convolution Definition Matrix 3 M¨obiusinversion 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 M¨obius 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 Theory, Lecture 3 Definition Jan Snellman An arithmetical function is a function f : P C. Arithmetical functions We will mostly deal with integer-valued a.f. Definition Some common ! arithmetical Euler φ is one: functions Dirichlet Convolution Matrix interpretation 25 Order, Norms, Infinite sums 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, M¨obiusfunction µ: Matrix interpretation Order, Norms, Infinite sums !(n) = r Multiplicative function Ω(n) = a1 + ··· + ar Definition Ω(n) Euler φ λ(n) = (-1) M¨obius inversion λ(n) !(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 kjn Order, Norms, X Infinite sums σ(n) = k Multiplicative function kjn Definition X Euler φ φ(n) = 1 M¨obius inversion 1≤k<n Multiplicativity is gcdX(k;n)=1 preserved by multiplication Matrix verification Divisor functions Euler φ again µ itself Number Even more Arithmetical functions Theory, Lecture 3 Jan Snellman p prime. Von 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 M¨obius 0 n ≡ 0 mod p inversion Multiplicativity is n 2 preserved by = 8+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 jn; pk+1 6 jn 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 M¨obius 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 Xkjn X`jn Infinite sums Multiplicative function Definition Euler φ Example M¨obius 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) M¨obius 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 2 and D(n) = f 1 ≤ k ≤ n kjn g 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 ijj, 0 otherwise Definition ij Euler φ • Similarly for a.f. g and matrix B M¨obius 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) = kjn 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 M¨obius 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 Xkjn µ itself 1<k≤n 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 ) = f n f (n) 6= 0 g Dirichlet Convolution Matrix interpretation and the norm Order, Norms, -ord(f ) Infinite sums kf k = 2 Multiplicative function Definition Euler φ Lemma M¨obius 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.