9 Arithmetic functions Definition 91 A function f : N C is called an arithmetic function Some examles of arithmetic functions include: 1 the identity function In { 1 if n 1, 0 if n > 1; 2 the constant function 1n 1; 3 the divisor function, dn #{d N : d n} sometimes denoted τn; 4 the Euler totient function ϕn #{a N : a n and a, n 1}; 5 the Möbius function, 1 if n 1, µn 1 r if n 1 r for distinct rimes i, 0 otherwise; 6 the von Mangoldt function Λn; 7 the function Nn n Definition 92 Let f, g be arithmetic functions Then their Dirichlet convolution is the arithmetic function f g defined by f gn fdgn/d For examle we have d 1 1 Let A denote the set of arithmetic functions It is straightforward to see A is a commutative ring with resect to Dirichlet convolution and the usual +, with identity element In In fact: Lemma 93 A is an integral domain Proof Exercise Show that f g g f and that A has no zero divisors Lemma 94 µ 1 I Proof We have µ 11 µ1 1 Let n > 1 with n a 1 1 ar r and 1 < < r Then µ 1n µd µ j r r 1 #J 1 k k J {1,,r} j J J k0 1 1 r This means that µ is the inverse of 1 under Dirichlet convolution As a simle corollary, we obtain: 36 0
Theorem 95 Möbius inversion Let f A and define gn fd Then fn gdµn/d Proof We have g f 1 if and only if f g µ Definition 96 An arithmetic function f has at most olynomial growth if there exists σ R such that fn On σ Let us denote by A oly the set of f A of at most olynomial growth Then one can show that A oly is a subring of A exercise For f A oly, the associated Dirichlet series fnn s defines an analytic function oome half lane Rs > σ + 1 This turns out to be a very useful device for understanding the ring structure of A oly : Theorem 97 Let f, g A oly Then f gn m1 Proof Exanding the right-hand side, we obtain fmgn mn s which is the left-hand side r1 mnr fn fmgn r s gn r1 f gr r s, In other words, the ma f fnn s is a ring homomorhism from A oly to the ring of functions that are analytic on a right half lane, and in fact this ma is injective exercise We have dn 1 1n ζs 2 Definition 98 An arithmetic function f is multilicative res comletely multilicative if fmn fmfn whenever m, n 1 res for all m, n If f A oly is multilicative and non zero then, generalising the roof of the Euler roduct formula for ζ, one finds that fn 1 + f + f2 + s 2s If f is comletely multilicative then the inner series is geometric, so that fn 1 1 f s Examle µ is multilicative Thus µn 1 + µ + µ2 + s 2s for Rs > 1 37 1 1 s 1 ζs,
Lemma 99 If f, g A are multilicative then f g is multilicative Proof Let m, n 1 Then where f gmn d mn d 1 fdgmn/d d 1 m m, r d d 2 n r and d 2 mn fd 1 d 2 g, d 1 d 2 n, r d Here r d means r d but r+1 d But then it follows from multilicativity that f gmn m n fd 1 fd 2 g g f gmf gn d 1 d 2 d 1 m d 2 n Remark If f A is not the zero function and f is multilicative then f1 1 Indeed, we have fn fn 1 fnf1 There are lots of identities between elements of A: Lemma 910 ϕ µ N where Nn n Proof We have ϕn 1 µd a n a n d a,n a n a,n1 since µ 1 I Switching the order of summation we get ϕn µd 1 µd n d a n, d a r µd, d a µ Nn It follows from this result that ϕ is multilicative since both µ and N are If n r is a rime ower then ϕ r r r 1 r 1 1 Hence it follows that ϕn n n 1 1 Alying Möbius inversion to ϕ µ N we deduce that ϕ 1 N; ie ϕd n for any n N Since ϕ µ N, we can easily calculate the Dirichlet series associated to the Euler totient function as ϕn µ Nn 38 µn 1 1 ζs 1 ζs
In articular we have σ a σ c 2 for this Dirichlet series Lemma 911 Λ 1 log Proof Recall that Λ1 0 and Λn Write n a 1 1 ar r Then Λd i r Next we claim that { log if n k, 0 otherwise Λ a i a i log i log n a a i Λn i r µd log d This obviously true for n 1 For n > 1, Möbius inversion gives Λn µd logn/d µd log n log d µd log d, as required, since µ 1 I More examles of arithmetic functions: 1 If n a 1 1 ar r with 1 < < r then ωn r and Ωn a 1 + + a r 2 The sum of divisors function is σ s n ds for s R Note that d σ 0 and one usually writes σ for σ 1 Returning to the divisor function, we have already seen that d 1 1 Hence Lemma 99 imlies that dn is a multilicative arithmetic function It is not comletely multilicative why? It is easy to see that 2 ωn dn 2 Ωn exercise Arithmetic functions can be quite erratically behaved and in the next section we will study their behaviour on average By multilicativity we have dn a 1 + 1 a r + 1 if n a 1 1 ar r In articular d 2 for all rimes, but sometimes dn can be much bigger: Lemma 912 Let k N Then dn log n k for infinitely many n N Proof Let 1 2, 2 3,, k+1 be the first k + 1 rimes Put n 1 2 k+1 m Then k+1 dn m + 1 k+1 > m k+1 log n log n k, log 1 2 k+1 if log n log 1 2 k+1 k+1 dn log n k Thus, roviding that m log 1 2 k+1 k, we have On the other hand dn can t be too big The following result shows, in articular, that the divisor function belongs to A oly Lemma 913 We have dn O ε n ε for any ε > 0 39
Proof Given ε > 0, we have to show that there is a ositive constant Cε such that dn Cεn ε for every n N By multilicativity we have dn n ε a n aε We decomose the roduct into two arts according to whether < 2 1/ε or 2 1/ε In the second art ε 2, so that 1 aε 2 a Thus we must estimate the first art Notice that aε 1 + a aε 1 + 1 ε log 2, since aε log 2 e aε log 2 2 aε aε Hence 1 + 1 Cε ε log 2 <2 1/ε 40