Problems on Chapter 1 15 is nonnegative for all real t, so that its discriminant is ≤ 0, that is, 4 k i=1 aibi 2 − 4 k i=1 ai 2 k i=1 bi 2 ≤ 0, which is precisely what needed to be proved. Problems on Chapter 1 Problem 1.1. Use Proposition 1.1 to prove that if α ≥ 0, then n≤x nα = xα+1 α + 1 + O (xα) . Problem 1.2. Prove that lim N→∞ 13 + 23 + 33 + 43 + · · · + N 3 N 4 = 1 4 . Problem 1.3. Use relation ( 1.1) to prove that if α 0, then n≤x nα log n = xα+1 α + 1 log x − 1 α + 1 + O (xα log x) . Problem 1.4. Show that the following two representations of the Euler constant γ are actually the same: lim N→∞ N n=1 1 n − log N and 1 − ∞ 1 t − t t2 dt. Problem 1.5. Let f : N → C be a function for which there exists a positive constant A such that lim x→∞ 1 x n≤x f(n) = A. Prove that n≤x f(n) log n = A(1 + o(1))x log x (x → ∞). Problem 1.6. Let f : [a, b] → R be a function which is continuous at x = a. Define g : [a, b] → R by g(x) = 0 if x = a, 1 if a x ≤ b. Prove that b a f dg = f(a).
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