The elementary theory of arithmetic functions
2.1 Mean values
The elementary theory of arithmetic functions
2.1 Mean values
We say that an arithmetic function F(n) has a mean value c if
N →∞lim 1 N
N n=1
F(n) = c.
In this section we develop a simple method by which mean values can be shown to exist in many interesting cases.
If two arithmetic functions f and F are related by the identity F(n) =
d|n
f(d), (2.1)
then we can write f in terms of F:
f(n) =
d|n
µ(d)F(n/d). (2.2)
This is the M¨obius inversion formula. Conversely, if (2.2) holds for all n then so also does (2.1). If f is generally small then F has an asymptotic mean value.
To see this, observe that
n≤x
F(n) =
n≤x
d|n
f(d).
By iterating the sums in the reverse order, we see that the above is
=
d≤x
f(d)
n≤xd|n
1 =
d≤x
f(d)[x/d].
35
36 The elementary theory of arithmetic functions
d=1 f(d)/d if this series converges and if
d≤x| f (d)| = o(x). This approach, though somewhat crude, often yields use-ful results.
Theorem 2.1 Let ϕ(n) be Euler’s totient function. Then for x ≥ 2,
prod-uct, we see that
ϕ(n)
proof is complete.
Let Q(x) denote the number of square-free integers not exceeding x, Q(x) =
n≤xµ(n)2. We now calculate the asymptotic density of these numbers.
Theorem 2.2 For all x ≥ 1, Q(x) = 6
π2x + O x1/2
.
Proof Every positive integer n is uniquely of the form n = ab2where a is square-free. Thus n is square-free if and only if b = 1, so that by (1.20)
d2|n
µ(d) =
d|b
µ(d) = µ(n)2. (2.4)
2.1 Mean values 37
This is a relation of the shape (2.1) where f (d) = µ(√
d) if d is a perfect square, and f (d) = 0 otherwise. Hence by (2.3),
Q(x) = x
d2≤x
µ(d) d2 + O
d2≤x
1
.
The error term is ≪ x1/2, and the sum in the main term is treated as in the
preceding proof.
We note that the argument above is routine once the appropriate identity (2.4) is established. This relation can be discovered by considering (2.2), or by using Dirichlet series: Let Q denote the class of square-free numbers. Then for σ >1,
n∈Q
n−s =
p
(1 + p−s) =
p
1 − p−2s
1 − p−s = ζ(s) ζ(2s).
Now 1/ζ (2s) can be written as a Dirichlet series in s, with coefficients f (n) = µ(d) if n = d2, f (n) = 0 otherwise. Hence the convolution equation (2.4) gives the coefficients of the product Dirichlet series ζ (s) · 1/ζ (2s).
Suppose that ak, bm, cnare joined by the convolution relation cn =
km=n
akbm, (2.5)
and that A(x), B(x), C(x) are their respective summatory functions. Then C(x) =
km≤x
akbm, (2.6)
and it is useful to note that this double sum can be iterated in various ways. On one hand we see that
C(x) =
k≤x
akB(x/k); (2.7)
this is the line of reasoning that led to (2.3) (take ak= f (k), bm = 1). At the opposite extreme,
C(x) =
m≤x
bmA(x/m), (2.8)
and between these we have the more general identity C(x) =
k≤y
akB(x/k) +
m≤x/y
bmA(x/m) − A(y)B(x/y) (2.9) for 0 < y ≤ x. This is obvious once it is observed that the first term on the right sums those terms akbmfor which km ≤ x, k ≤ y, the second sum includes the
38 The elementary theory of arithmetic functions
pairs (k, m) for which km ≤ x, m ≤ x/y, and the third term subtracts those akbm
for which k ≤ y, m ≤ x/y, since these (k, m) were included in both the previous terms. The advantage of (2.9) over (2.7) is that the number of terms is reduced (≪ y + x/y instead of ≪ x), and at the same time A and B are evaluated only at large values of the argument, so that asymptotic formulæ for these quantities may be expected to be more accurate. For example, if we wish to estimate the average size of d(n) we take ak= bm= 1, and then from (2.3) we see that
n≤x
d(n) = x log x + O(x).
To obtain a more accurate estimate we observe that the first term on the right-hand side of (2.9) is
k≤y
[x/k] = x
k≤y
1/k + O(y).
By Corollary 1.15 this is
xlog y + C0x + O(x/y + y).
Here the error term is minimized by taking y = x1/2. The second term on the right in (2.9) is then identical to the first, and the third term is [x1/2]2= x + O(x1/2), and we have
Theorem 2.3 For x ≥ 2.
n≤x
d(n) = x log x + (2C0− 1)x + O x1/2
.
We often construct estimates with one or more parameters, and then choose values of the parameters to optimize the result. The instance above is typical – we minimized x/y + y by taking y = x1/2. Suppose, more generally, that we wish to minimize T1(y) + T2(y) where T1 is a decreasing function, and T2 is an increasing function. We could differentiate and solve for a root of T1′(y) + T2′(y) = 0, but there is a quicker method: Find y0so that T1(y0) = T2(y0). This does not necessarily yield the exact minimum value of T1(y) + T2(y), but it is easy to see that
T1(y0) ≤ miny (T1(y) + T2(y)) ≤ 2T1(y0),
so the bound obtained in this way is at most twice the optimal bound.
Despite the great power of analytic techniques, the ‘method of the hyperbola’
used above is a valuable tool. The sequence cn given by (2.5) is called the Dirichlet convolutionof akand bm; in symbols, c = a ∗ b. Arithmetic functions form a ring when equipped with pointwise addition, (a + b)n= an+ bn, and
2.1 Mean values 39
Dirichlet convolution for multiplication. This ring is called the ring of formal Dirichlet series. Manipulations of arithmetic functions in this way correspond to manipulations of Dirichlet series without regard to convergence. This is analogous to the ring of formal power series, in which multiplication is provided by Cauchy convolution, cn =
k+m=nakbm.
In the ring of formal Dirichlet series we let O denote the arithmetic function that is identically 0; this is the additive identity. The multiplicative identity is i where i1= 1, in= 0 for n > 1. The arithmetic function that is identically 1 we denote by 1, and we similarly abbreviate µ(n), (n), and log n by µ, Λ, and L. In this notation, the characteristic property of µ(n) is that µ ∗ 1 = i, which is to say that µ and 1 are convolution inverses of each other, and the M¨obius inversion formula takes the compact form
a ∗ 1 = b ⇐⇒ a = b ∗ µ.
In the elementary study of prime numbers the relations Λ ∗ 1 = L, L ∗ µ = Λ are fundamental.
2.1.1 Exercises
1. (de la Vall´ee Poussin 1898; cf. Landau 1911) Show that
n≤x
{x/n} = (1 − C0)x + O x1/2
where C0is Euler’s constant, and {u} = u − [u] is the fractional part of u.
2. (Duncan 1965; cf. Rogers 1964, Orr 1969) Let Q(x) be defined as in The-orem 2.2.
(a) Show that Q(N ) ≥ N −
p[N / p2] for every positive integer N . (b) Justify the relations
p
1 p2 < 1
4+
∞ k=1
1
(2k + 1)2 < 1 4+1
2
∞ k=1
1 2k − 1
2k + 2
= 1/2.
(c) Show that Q(N ) > N /2 for all positive integers N .
(d) Show that every positive integer n > 1 can be written as a sum of two square-free numbers.
3. (Linfoot & Evelyn 1929) Let Qkdenote the set of positive kthpower free integers (i.e., q ∈ Qkif and only if mk|q ⇒ m = 1).
(a) Show that
n∈Qk
n−s = ζ(s) ζ(ks) for σ > 1.
40 The elementary theory of arithmetic functions
(b) Show that for any fixed integer k > 1
4. (cf. Evelyn & Linfoot 1930) Let N be a positive integer, and suppose that Pis square-free.
(d) Take P to be the product of all primes not exceeding y. By letting y tend to infinity slowly, show that the number of ways of writing N as a sum of two square-free integers is ∼ c(N )N where
c(N ) = a 5. (cf. Hille 1937) Suppose that f (x) and F(x) are complex-valued functions
defined on [1, ∞). Show that
F(x) =
6. (cf. Hartman & Wintner 1947) Suppose that
| f (n)|d(n) < ∞, and that
2.1 Mean values 41
for all n if and only if
f(n) =
m n|m
µ(m/n)F(m).
7. (Jarn´ık 1926; cf. Bombieri & Pila 1989) Let C be a simple closed curve in the plane, of arc length L. Show that the number of ‘lattice points’ (m, n), m, n ∈ Z, lying on C is at most L + 1. Show that if C is strictly convex then the number of lattice points on C is ≪ 1 + L2/3, and that this estimate is best possible.
8. Let C be a simple closed curve in the plane, of arc length L that encloses a region of area A. Let N be the number of lattice points inside C. Show that |N − A| ≤ 3(L + 1).
9. Let r (n) be the number of pairs ( j, k) of integers such that j2+ k2= n.
Show that
n≤x
r(n) = π x + O x1/2
. 10. (Stieltjes 1887) Suppose that
an,
bn are convergent series, and that cn =
km=nakbm. Show that
cnn−1/2converges. (Hence if two Dirichlet series have abscissa of convergence ≤ σ then the product series γ (s) = α(s)β(s) has abscissa of convergence σc≤ σ + 1/2.)
11. (a) Show that
n≤xϕ(n) = (3/π2)x2+ O(x log x) for x ≥ 2.
(b) Show that
n≤xm≤x
(m,n)=1
1 = −1 + 2
n≤x
ϕ(n)
for x ≥ 1. Deduce that the expression above is (6/π2)x2+ O(x log x).
12. Let σ (n) =
d|nd. Show that
n≤x
σ(n) =π2
12x2+ O(x log x) for x ≥ 2.
13. (Landau 1900, 1936; cf. Sitaramachandrarao 1982, 1985, Nowak 1989) (a) Show that n/ϕ(n) =
d|nµ(d)2/ϕ(d).
(b) Show that
n≤x
n
ϕ(n) = ζ(2)ζ (3)
ζ(6) x + O(log x) for x ≥ 2.
42 The elementary theory of arithmetic functions
14. Let κ be a fixed real number. Show that
15. (cf. Grosswald 1956, Bateman1957)
(a) By using Euler products, or otherwise, show that 2ω(n)=
(c) Show also that
16. (a) Show that for any positive integer q,
(b) Show that for any real number x ≥ 1 and any positive integer q,
2.1 Mean values 43
(c) Show that for any real number x ≥ 2 and any positive integer q,
(c) Show that for every fixed positive integer k,
19. (cf. Erd˝os & Szekeres 1934, Schmidt 1967/68) Let Andenote the number of non-isomorphic Abelian groups of order n.
(a) Show that∞
20. (Wintner 1944, p. 46) Suppose that
d|g(d)|/d < ∞. Show that
d≤x|g(d)| = o(x). Suppose also that
n≤x f(n) = cx + o(x), and put h(n) =
d|n f(d)g(n/d). Show that
44 The elementary theory of arithmetic functions
embarrassing that this is the best-known upper bound for gaps between sums of two squares.)
22. (Feller & Tornier 1932) Let f (n) denote the multiplicative function such that f ( p) = 1 for all p, and f ( pk) = −1 whenever k > 1.
(a) Show that
∞ n=1
f(n)
ns = ζ (s)
p
1 − 2
p2s
for σ > 1.
(b) Deduce that
f(n) =
d2|n
µ(d)2ω(d).
(c) Explain why 2ω(n)≤ d(n) for all n.
(d) Show that
n≤x
f(n) = ax + O
x1/2log x where a is the constant of Exercise 3.
(e) Let g(n) denote the number of primes p such that p2|n. Show that the set of n for which g(n) is even has asymptotic density (1 + a)/2.
(f) Put
ek= 1 k
d|k
µ(d)2k/d. Show that if |z| < 1, then
log(1 − 2z) =
∞ k=1
eklog 1 − zk
.
(g) Deduce that
a = ∞ k=1
ζ(2k)ek.
Note that the kth factor here differs from 1 by an amount that is
≪ 1/(k2k). Hence the product converges very rapidly. Since ζ (2k) can be calculated very accurately by the Euler–Maclaurin formula (see Appendix B), the formula above permits the rapid calculation of the constant a.
2.1 Mean values 45
46 The elementary theory of arithmetic functions
(c) Suppose that Q ≥ 1 is an integer, B ≥ 1, and that 1/Q3≤ ± f′′(x) ≤ B/Q3 for 0 ≤ x ≤ N where the choice of sign is independent of x. Show that numbers ar, qr, Nr can be determined, 0 ≤ r ≤ R for some R, so that (i) (ar,qr) = 1, (ii) qr ≤ Q, (iii) | f′(Nr) − ar/qr| ≤ 1/(qrQ), and (iv) N0= 0, Nr = Nr −1+ qr −1for 1 ≤ r ≤ R, N − Q ≤ NR ≤ N .
(d) Show that under the above hypotheses
N n=0
B1({ f (n)}) ≪ B(R + 1) + Q.
(e) Show that the number of s for which as/qs= ar/qr is ≪ Q2/q2. Let 1 ≤ q ≤ Q. Show that the number of r for which qr = q is
≪ (Q/q)2(B N q/Q3+ 1).
(f) Conclude that under the hypotheses of (c),
N n=0
B1({ f (n)}) ≪ B2N Q−1log 2Q + B Q2.
26. Show that if U ≤√ x, then
U <n≤2U
B1({x/n}) ≪ x1/3log x.
Let (x) be as in Exercise 23(b). Show that (x) ≪ x1/3(log x)2. 27. Let R(x) be as in Exercise 24(c). Show that R(x) ≪ x1/3log x.