LECTURE 11: ARITHMETIC FUNCTIONS Recall that a function f : N ! C is called an arithmetical function. Re- call also that a multiplicative function f satisfies the property that whenever (m; n) = 1, one has f(mn) = f(m)f(n): We have proved that the function X g(n) = f(d) djn is multiplicative whenever f(n) is multiplicative. In this section we discuss various properties of arithmetical functions, many of them multiplicative, and seek to understand what they \look" like. 1. Examples of arithmetical functions • the divisor function τ(n). The function τ(n) is defined as the number of divisors of the number n. It can be written explicitly as X τ(n) = 1: djn • the sum of divisors function σ(n). The function σ(n) is defined as the sum of all divisors of the number n. It can be written explicitly as X σ(n) = d: djn • the Euler totient function φ(n). The function φ(n) is defined as the number of reduced residue classes modulo n. We have shown that it is given by a formula Y φ(n) = (pr − pr−1): prkn • the M¨obiusfunction µ(n) The function φ(n) is defined by ( (−1)`; when n = p ··· p is squarefree, µ(n) = 1 ` 0; otherwise. Here, by a squarefree number, we mean an integer that is not divisible by the square of any prime number. It is easy to check that µ is multiplicative. 1 2 LECTURE 11 Note that, just as with our earlier discussion of the Euler totient, a function that is multiplicative will be relatively easy to evaluate when its argument has a known prime factorisation. For example, one can see rather easily that when p is a prime number and h is a non-negative integer, then h X pr+1 − 1 τ(pr) = r + 1 and σ(pr) = pi = ; p − 1 i=0 and thus Y Y pr+1 − 1 τ(n) = (r + 1) and σ(n) = : p − 1 prkn prkn 2. The Mobius¨ inversion formula The M¨obiusfunction, has special properties that make it particularly useful in studying averages of other arithmetic functions (and much else besides). Recall that ( (−1)`; when n = p : : : p is squarefree, µ(n) = 1 ` 0; otherwise. We define a rather trivial multiplicative function ν(n) by ( 1; when n = 1, ν(n) = 0; otherwise. P Lemma 2.1. One has djn µ(d) = ν(n). P Proof. Since µ(n) is a multiplicative function of n, it follows that djn µ(d) is P also multiplicative. But on writing f(n) = djn µ(d), one finds that α X f(pα) = µ(ph) = 1 − 1 = 0; for α > 0; h=0 and f(1) = µ(1) = 1. Thus, in view of the multiplicativity of f(n), one finds that f(n) is zero unless n has no prime divisors, a circumstance that occurs only when n = 1. This completes the proof of the theorem. We can now describe a certain duality between arithmetic functions, and functions defined via divisor sums. Theorem 2.2 (the M¨obiusinversion formulae). (i) Let f be any arithmetical function, and define X g(n) = f(d): djn Then one has X f(n) = µ(d)g(n=d): djn LECTURE 11 3 (ii) Suppose that g is any arithmetical function, and define X f(n) = µ(d)g(n=d): djn Then one has X g(n) = f(d): djn P Proof. (i) Given that g(n) = djn f(d), one obtains X X X X X µ(d)g(n=d) = µ(d)f(e) = f(e) µ(d) djn djn ej(n=d) ejn dj(n=e) X = f(e)ν(n=e) = f(n): ejn P (ii) Given that f(n) = djn µ(d)g(n=d), one obtains X X X X f(d) = f(n=d) = µ(e)g(n=(de)) djn djn djn ej(n=d) X X X X = µ(e)g(n=(de)) = µ(e)g(d) ejn dj(n=e) ejn dj(n=e) X X X = g(d) µ(e) = g(d)ν(n=d) = g(n): djn ej(n=d) djn Note that M¨obiusinversion applies to all arithmetical functions, without any hypothesis concerning whether or not they are multiplicative. Example 2.3. Recall that we showed that X φ(d) = n: djn As an immediate consequence of the M¨obiusinversion formulae, we deduce that X X φ(n) = µ(d)(n=d) = n µ(d)=d: djn djn 3. Estimates for arithmetical functions We now explore the \population statistics" of values of arithmetical func- tions: what is the maximal/minimal size of such a function, the average size, the variance, etc.? In order properly to discuss such issues, we need to recall some standard analytic notation. Given functions f; g : R ! R, with g taking positive values, we write f(x) = O(g(x)) (for x > x0) 4 LECTURE 11 when there exists some positive constant C for which jf(x)j 6 Cg(x) (for x > x0). 2 2 Example 3.1. One has x = O(x ) for x > 1, 1=x = O(1) for x > 1, and x x = O(e ) for x > 0. There are two useful strategies to keep in mind when addressing questions concerning estimates for arithmetic functions: • In order to estimate the size of a multiplicative function f(n), one should first estimate f(pr) on prime powers, and then combine this information with knowledge about the distribution of prime numbers. • If one wishes to estimate the average size of an arithmetical function g(n), one can apply the M¨obiusinversion formulae to write g(n) in the shape X g(n) = f(d); djn in which X f(n) = µ(d)g(n=d): djn Frequently, one finds that this new function f(n) is reasonably well behaved, and then one has X X X g(n) = f(d) 16n6x 16n6x djn X X = f(d): 16d6x 16m6x=d Here, in the last summation, we made the change of variable n = md. Thus we obtain X X X g(n) = f(d) 1 16n6x 16d6x 16m6x=d X jxk = f(d) ; d 16d6x where, as usual, we write bθc for the greatest integer not exceeding θ. Thus we see that 1 X 1 X x g(n) = f(d) + O(1) x x d 16n6x 16d6x ! X f(d) 1 X = + O jf(d)j : d x 16d6x 16d6x In many circumstances, the first term on the right hand side of the last equation is of larger order of magnitude than the last term, and then LECTURE 11 5 one has the asymptotic formula 1 X X f(d) g(n) ∼ : x d 16n6x 16d6x This formula is useful provided that the new average is easier to com- pute than the original average. We illustrate these ideas with some examples. Example 3.2. The divisor function τ(n). We claim that for any positive number ", one has " τ(n) = O(n ) for n 2 N. In order to establish this estimate, we exploit the multiplicative property of τ(n), and investigate the function τ(n) Y j + 1 = : n" pj" pj kn For any fixed prime p,(j + 1)=pj" is a decreasing function of j for sufficiently large j. In fact, with a little calculus we find that " " j + 1 p (j + 1)" log2 p p −x " j" = (j+1)" log p 6 supfx2 : x > 0g = O(p ): p " log2 p 2 2 " log2 p Thus, τ(n) Y O(p") = O(n"): n" 6 pj kn Next we turn to the matter of the average value of τ(n) for 1 6 n 6 x. In P this instance, of course, one has τ(n) = djn 1, and so X X X X X τ(n) = 1 = 1 16n6x 16n6x djn 16d6x 16m6x=d X X = bx=dc = (x=d + O(1)) 16d6x 16d6x X 1 = x + O(x): d 16d6x But (as a good exercise in calculus), X 1 X Z d+1 dt = + O(1=d2) d d t 16d6x 16d6x = log x + O(1): Thus we deduce that 1 X τ(n) = log x + O(1): x 16n6x Example 3.3. The sum of divisors function σ(n). 6 LECTURE 11 P In this instance we have the formula σ(n) = djn d, and so X X X n X X σ(n) = = m d 16n6x 16n6x djn 16d6x 16m6x=d X X x2 = 1 bx=dc(bx=dc + 1) = + O(x=d) : 2 2d2 16d6x 16d6x But 1 ! X 1 X 1 X = + O 1=d2 = ζ(2) + O(1=x); d2 d2 16d6x d=1 d>x and hence 1 X π2 σ(n) = 1 ζ(2)x + O(log x) = x + O(log x): x 2 12 16n6x Example 3.4. The Euler function φ(n). We shall use the formula X µ(d) φ(n) = n : d djn Following the above strategy, we find that X X X X X φ(n) = µ(d)n=d = µ(d) m 16n6x 16n6x djn 16d6x 16m6x=d X X x2 = µ(d) · 1 bx=dc(bx=dc + 1) = µ(d) + O(x=d) 2 2d2 16d6x 16d6x ! X µ(d) X = 1 x2 + O jµ(d)jx=d : 2 d2 16d6x 16d6x To estimate the error term, we observe that X 1 Z x dt < 1 + = O(log x): d 1 t 16d6x Furthermore, 1 ! 1 X µ(d) X µ(d) X X µ(d) = + O 1=d2 = + O(1=x): d2 d2 d2 16d6x d=1 d>x d=1 Thus 1 X φ(n) = Cx + O(log x); x 16n6x P1 µ(d) 2 where C = d=1 d2 .