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).
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2012 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.