Exercises 37

10. (Ramanujan) Assuming ζ(2) =

π2/6

and ζ(4) =

π4/90,

show that

1

n2

=

9

2π2

,

where the indicates that the sum ranges over positive squarefree inte-

gers n with an odd number of prime divisors.

11. (Cf. Porubsk´ y [Por01]) If R is a commutative ring, its Jacobson radical

J(R) is the intersection of all of its maximal ideals. Show that

J(R) = {x ∈ R : 1 − xy is invertible for all y ∈ R}.

Deduce that if R is an integral domain with finitely many units, then

J(R) = {0}. Use this to prove that if R is a principal ideal domain with

finitely many units, then either R is a field or R contains an infinite set

of pairwise nonassociated primes.

12. By carefully examining the proof of Theorem 1.10, show that the the-

orem remains correct when A = 1, provided that in condition (ii) we

replace “prime” with “prime or equal to 1”.

13. Suppose that a1 a2 a3∑ · · · is an increasing sequence of natural

numbers, and put A(x) :=

ai≤x

1. Prove that if (log

x)−kA(x)

→ ∞

for each fixed k, then infinitely many primes p divide some ai. Use this

to give another proof of Lemma 1.17.

14. Prove the following theorem of Bauer [Bau06]:

Theorem. If F (T) ∈ Z[T] is a nonconstant polynomial with at least

one real root, then for every m ≥ 3, there exist infinitely many prime

divisors p of F with p ≡ 1 (mod m).

Proceed by showing that each of the following conditions on F is

suﬃcient for the conclusion of the theorem to hold:

(a) F has a positive leading coeﬃcient and constant term −1.

(b) F has a positive leading coeﬃcient and negative constant term.

(c) F has a positive leading coeﬃcient and F (a) 0 for some a ∈ Z.

(d) F has a positive leading coeﬃcient and F (a) 0 for some a ∈ Q.

(e) F has a positive leading coeﬃcient and F (a) 0 for some a ∈ R.

(f) F has a positive leading coeﬃcient and F (a) = 0 for some a ∈ R.

Hint for (f): Reduce to the case when F has no multiple roots.

15. Let F be a field of characteristic not dividing m. By carefully examining

the proof of Lemma 1.19, show that the roots of Φm(T) in the algebraic

closure of F are precisely the primitive mth roots of unity there, and

that all these roots are simple.