Download Math 675, Homework 4, Part 1 (Due Monday, October 26, 2015, in

Math 675, Homework 4, Part 1 (Due Monday, October 26, 2015, in class)
(or by 3pm in the box outside EH3086)
Problems with a star, if there are any, might be harder. Two stars are not assigned.
29. (Squarefull numbers) Let S = {n ≥ 1 : p|n implies p2 |n}. Let S(x) = #{n ∈
S : n ≤ x}.
(a) Show that each n ∈ S can be written in a unique way as n = m3 d2 where m is
squarefree.
p
P
(b) Estimate T (x) := n≤x,n∈S nx and use this to deduce that
S(x) = O
√
x(log x)
(c) Use (a) and (b) to show that
ζ( 23 ) √
S(x) ∼
x,
ζ(3)
as x → ∞.
Q
30. (Radical of an integer) The radical of an integer n is Rad(n) = p|n p, which is the
largest squarefree divisor of n. It is also called the squarefree kernel of n.
(a) Use the fact that Rad(n)/n is a multiplicative function of size at most 1 to deduce
that
X Rad(n)
∼ Cx as x → ∞,
n
n≤x
with constant C given by
C :=
Y
1−
p
1
.
p(p + 1)
(b) Use the result of Problem 29 (c), to show that
X Rad(n)
n≤x
n
= Cx + O
√ x .
(c) Deduce from (b) that
1
Rad(n) = Cx2 + O x3/2 .
2
n≤x
X
31. Let ω(n) count the number of prime divisors of n without multiplicity, Ω(n) count
number of prime divisors counted with multiplicity, and d(n) the number of divisors of
n.
(a) Show that there is a constant A such that
x X
ω(d(n)) = Ax + O
.
log x
n≤x
1
(b) Show that
X
Ω(d(n)) = x log log x + O x .
n≤x
(Compare these asymptotics with those of Problem 25.)
32. (Integers with no medium-sized prime factors) Let x be given and for 1 ≤ y ≤ z ≤ x,
let Ψ0 (x, y, z) count the number of integers below x which have no prime divisor in the
interval y < p ≤ z.
(a) For fixed y, z, letting x vary, use the Eratosthenes sieve (inclusion-exclusion) to
show that
Y 1
1
1−
lim Ψ0 (x, y, z) =
x→∞ x
p
y<p≤z
(b) Improve this estimate further using Brun’s pure sieve method to show that
there is a positive absolute constant c, such that one has, uniformly for 1 ≤ y ≤ z ≤
exp(c log x/(log log x)), the asymptotic estimate
Y 1
Ψ0 (x, y, z) ∼ x
1−
, as x → ∞.
p
y<p≤z
(c) Show, assuming the prime number
theorem, that the estimate of (b) cannot hold
√
uniformly in the range 1 ≤ y < z ≤ x.
(d) Further improve the range of z allowed in (c) using the Fundamental Lemma of
combinatorial sieve theory (handout) as a black box. Note that (d) provides a limit on
how far one can get.
33. (Primes of the form n2 +1) It is not known whether there are infinitely many primes
of the form n2 + 1. Here we obtain an upper bound on their number.
(a) Show that the number ρ(p) of solutions of the equation x2 + 1 ≡ 0 (mod p) has
n 1
ρ(p) = 2
0
if p = 2,
if p ≡ 1 (mod 4),
if p ≡ 3 (mod 4).
2
Q (b) Deduce that the equation x + 1 ≡ 0 mod d) has, for each squarefree d, ρ(d) :=
p|d ρ(p) solutions.
(c) Using the Fundamental Lemma of Combinatorial Sieve Theory (handout) as a
black box, show that the number S(x) of primes ≤ x that are of the form n2 + 1 satisfies
√
Y
2 (1 − )
S(x) = O x
p
p≤x
p≡1( mod
4)
(d) Now assume as known Dirichlet’s theorem, stating that
X
1
1
= log log x + O 1 .
p
2
p≤x
p≡1( mod
4)
Apply it to simplify the second term in the estimate in (c) as a function of x.
2