1.4. Stieltjes integrals 7 = 1 + O 1 x + x 1 dt t − x 1 t − t t2 dt = 1 + O 1 x + log t t=x t=1 − ∞ 1 t − t t2 dt − ∞ x t − t t2 dt = log x + γ + O 1 x + ∞ x t − t t2 dt = log x + γ + O 1 x + O ∞ x dt t2 = log x + γ + O 1 x , as x → ∞, thus completing the proof of the theorem. 1.4. Stieltjes integrals At times, it is convenient to write certain finite sums as Stieltjes integrals. Recall that the Stieltjes1 integral I of the function f over the interval [a, b] with respect to the function g is defined as I = b a f dg = b a f(t) dg(t) = lim Δ→0 n j=1 f(ξj)(g(tj) − g(tj−1) with Δ = (t0,...,tn), a = t0, b = tn, Δ = max1≤j≤n(tj − tj−1) and ξj ∈ [tj−1,tj] for all j = 1,...,n. If g(x) = x, then I = b a f(t) d t is simply the Riemann integral of f. If g(x) = 0 if a ≤ x ≤ c, 1 if c x ≤ b, then I = f(c). For instance, assume that we want to estimate the expression (1.6) a≤x a∈A f(a), 1Thomas Stieltjes (1850–1894, Holland) was very much interested in elliptic curves and in number theory. In fact, as Narkiewicz recalls in his book [110], Stieltjes was the first to make an attempt at a proof of the Riemann Hypothesis. Indeed, he asserted that the series ∞ n=1 μ(n) ns converges for all s 1/2, a statement which would have implied the Riemann Hypothesis (see Problem 3.13 in Chapter 3).

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