Math 539
Math 539 Homework #1
due Monday, September 26, 2005 at 10 AM
Reality check problems. Not to write up; just ensure that you know how to do them.
I. Bateman and Diamond, p. 16, #2.3
II. Bateman and Diamond, p. 32, #2.16
III. Bateman and Diamond, p. 35, #2.18
IV. Justify the chain of reasoning
∑
p
∞
1
1
1
1 1 ∞
+
+
<
<
2
4 k∑
4 2 k∑
p2
=1 (2k + 1 )
=1
1
1
−
2k 2k + 2
1
= .
2
V. Let f and g be functions defined on the interval (0, ∞) that take values in (0, ∞)
as well. Suppose that c(t) is an increasing continuous function such that
lim c(t) = 0
and
t→0+
lim c(t) = ∞.
t→∞
Prove that f ( x) g( x) if and only if f (c(t)) g(c(t)).
Homework problems. To write up and hand in.
I. (a) Show that 2 x = 1 + O( x) for x in any bounded range.
(b) What is wrong with the following “proof” that 2 N N 2 ?
2N = 2 + 2N − 2 = 2 +
∑
2n
n< N
= O(1) +
∑ (1 + O(n))
n< N
= O(1) +
∑
n< N
1+O
∑
n
n< N
N ( N − 1)
= O(1) + N − 1 + O
2
= O( N 2 ).
II. Let Q( x) denote the number of squarefree integers not exceeding x. We know that
Q( x) ∼ π62 x, but the following elementary argument has merit as well.
(a) Prove that Q( N ) ≥ N − ∑ p b N / p2 c for every positive integer N. Conclude
that Q( N ) > N /2 for every N.
(b) Prove that every integer greater than 1 can be written as the sum of two
squarefree numbers.
III. For this problem, all functions are defined on the interval (0, ∞) and take values
in (0, ∞) as well (other than log).
(a) Suppose that a( y) is an increasing concave function. Prove that h( x) k( x)
implies a(h( x)) a(k( x)). Show via a counterexample that the concavity
hypothesis is necessary.
(b) Suppose that b( y) is an increasing convex function. Prove that b( f ( x)) b( g( x)) implies f ( x) g( x). Show via a counterexample that the convexity
hypothesis is necessary.
(c) For all real numbers A > 0 and 0 < b < 1 and ε > 0, show that
log A x exp(logb x) xε
uniformly for x ≥ 1.
IV. (a) Prove the “Cauchy Condensation Test”: if { an } is a decreasing sequence of
positive numbers, then the two series
∞
∑ an
∞
and
n=1
∑ 2 n a 2n
n=1
either both converge or both diverge.
(b) Let logk n denote the k-fold iterated logarmithm log log · · · log n. Prove that
for every k ≥ 1 and every ε > 0,
∑
n : logk
1
<∞
n(log n)(log log n) · · · (logk−1 n)(logk n)1+ε
n>0
while
∑
n : logk
1
= ∞.
n(log n)(log log n) · · · (logk−1 n)(logk n)1−ε
n>0
V. Let k > 1 be a fixed squarefree integer. Are integers that are divisible by k more or
less likely to be squarefree than those that are not divisible by k? Turn this into a
rigorous question and resolve it.
VI. (a) In one of my papers, I have as a lemma the inequality
2ω(n) ≤ 2
log 16
4− log 11
log 2
n log 11
for all n ∈ N. Prove this lemma (can you improve the constant?) and generalize.
(b) Prove that τ (n) ε nε for all ε > 0. Why is it impossible for τ (n) nε
uniformly for all ε > 0?
(c) Find the smallest constant C such that τ (n) ≤ Cn1/4 for all n ∈ N.
VII. Let ν p (n) denote the exponent of p in the prime factorization of n, so that there
exists an integer m with p - m and n = pν p (n) m. Compute the average value of
ν p (n); of ν p (n)2 . Is the second answer the square of the first? Should it be?
VIII. (a) What is wrong with the following beginning of an attempt to find the average
order of ω2 ?
2
x
2
∑ ω(n) = ∑ ∑ 1 = ∑ ∑ ∑ 1 = ∑ ∑ ∑ 1 = ∑ ∑ p1 p2 .
p1 ≤ x p2 ≤ x
n≤ x p |n p |n
p1 ≤ x p2 ≤ x n≤ x
n≤ x
n≤ x
p|n
1
2
p1 p2 |n
(b) Correct this beginning and carry through the evaluation of the average order
of ω2 .
D EFINITION . The (natural) density δ ( S) of a set S of positive integers is the average
value of its indicator function, that is,
1
1
δ ( S) = lim #{n ≤ x : n ∈ S} = lim
∑ 1,
x→∞ x
x→∞ x
n≤ x
n∈ S
if the limit exists (if it doesn’t exist, we say the set does not have a density). For
example, the set of odd integers has density 21 , and the set of primes has density 0.
IX. (a) Pick a number between 1 and 9 (inclusive).
(b) Show that the set of positive integers whose first digit is the number you chose
in part (a) does not have a density.
X. Let f be an arithmetic function that takes only finitely many values v1 , . . . , vm .
Suppose that for each 1 ≤ j ≤ m, the density δ j of the set {n ∈ N : f (n) = v j }
exists.
(a) Prove that ∑m
j=1 δ j = 1.
(b) Prove that the average value
1
α
Aα = lim
∑ f (n)
x→∞ x
n≤ x
exists for any α ∈ R (with the stipulation that none of the v j equals 0 if α < 0).
(c) Suppose the values are indexed so that δ1 ≥ δ2 ≥ · · · ≥ δm . Show that if the
average values A1 , A2 , . . . , Am are known, then all of the densities δ1 , . . . , δm
can be recovered.