6 1. Preliminary Notions = A(N)f(N) − N 1 A(t)f (t)dt = n≤N anf(n), where the last equality holds since we just proved that (1.4) holds when x is an integer. Now, since clearly ∑ n≤x anf(n) = ∑ n≤N anf(n), the proof of the proposition is complete. Before establishing the next result, we introduce an important constant. Definition 1.5. Let γ = 1 − ∞ 1 t − t t2 dt. The number γ is called Euler’s constant. Its numerical value is approxi- mately 0.57721 . . . . Remark 1.6. One can easily show that the above definition of the Eu- ler constant is equivalent to the following one (already mentioned in the Notation section on page xiii): γ = lim N→∞ N n=1 1 n − log N (see Problem 1.4). It is believed that γ is a transcendental number although it is not even known if it is irrational. See J. Havil’s excellent book [78] for a thorough study of this amazing constant. As an application of this formula we have the following. Theorem 1.7. For all x ≥ 1, (1.5) n≤x 1 n = log x + γ + O 1 x . Proof. Setting an = 1 and f(t) = 1/t in the Abel summation formula, we easily obtain that A(x) = n≤x 1 = x , and that f (t) = −1/t2, yielding n≤x 1 n = x x + x 1 t t2 dt = x − (x − x ) x + x 1 t − (t − t ) t2 dt

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