36 1. Elementary Prime Number Theory, I (d) If e and r are nonnegative integers and r is even, show that there is an expansion of the form dr dxr (P (x)e) = r j=r/2 cjj! e j P (x)e−j for certain integers cj. (e) Use the result of part (d) to show that if e is a nonnegative integer and r ≥ 2n is even, then 1 n! dr dxr (P (x)e) is a polynomial in P (x) with integer coeﬃcients. Conclude that f (r) (0) and f (r) (π) are integers. (f) Referring back to definition (1.16), deduce that F (π) + F (0) ∈ Z. 7. In this exercise we present a proof similar to that of J. Hacks (on p. 8) but relying on the irrationality of π in place of π2. Let χ(n) = (−1)(n−1)/2 if 2 n, 0 otherwise. a) Show that χ(n) is a completely multiplicative function, i.e., χ(ab) = χ(a)χ(b) for every pair of positive integers a, b. b) Assume that there are only finitely many primes. Show that for every s 0, ∞ n=1 χ(n) ns = p 1 − χ(p) ps −1 . c) Take s = 1 and obtain a contradiction to the irrationality of π. You may assume that π 4 = 1 − 1 3 + 1 5 − 1 7 + · · · . 8. Say that a natural number n is squarefull if p2 | n whenever p | n, i.e., if every prime showing up in the factorization of n occurs with multi- plicity 1. Every perfect power is squarefull, but there are many other examples, such as 864 = 25 · 33. Using Theorem 1.2, show that ∑ n−1 converges to ζ(2)ζ(3) ζ(6) , where the indicates that the sum is restricted to squarefull n. Determine the set of real α for which ∑ n−α converges. 9. (Continuation) Show that every squarefull number has a unique repre- sentation in the form u2v3, where u and v are positive integers with v squarefree. Deduce that for x ≥ 1, n≤x n squarefull 1 = ζ(3/2) ζ(3) x1/2 + O(x1/3).

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