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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2009 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.