4 1. Arithmetic Functions into T(x) − 2 T(x/2) yields ψ(x) − ψ x 2 + ψ x 3 − · · · = T(x) − 2 T x 2 . Then ψ(x) ≥ log(2)x − log(4x) since ψ is an increasing and nonnegative function, and ψ(x) − ψ x 2 ≤ log(2)x + log(x) for the same reason. Adding up the inequalities ψ x 2j − ψ x 2j+1 ≤ log(2)2−jx + log(x) for j = 0, 1, 2,..., [log(x)/ log(2)] − 1 yields ψ(x) ≤ 2 log(2)x + log(x) log(2) log(x) ≤ 2 log(2)x + log2(x) log(2) since ψ(x/2j+1) = 0 when x/2j+1 2. The inequalities in Proposition 1.1 and the limits log2(x)/(x log(2)) → 0 and log(4x)/x → 0 as x → +∞ imply that for every ε 0 there exists some x0(ε) so that log(2) − ε ψ(x)/x 2 log(2) + ε for x ≥ x0(ε). Such ε- x0(ε) inequalities are often expressed in a somewhat different but equivalent way, using concepts from analysis. The limit superior lim sup x→+∞ f(x) of a bounded real function f(x) on an interval [a, ∞) is the unique real number σ such that f(x) σ+ε holds for all x suﬃciently large, while f(x) σ−ε fails for some x arbitrarily large, no matter how small ε 0 is taken. Similarly the limit inferior lim infx→+∞ f(x) is the unique real number ι such that f(x) ι − ε holds for all x suﬃciently large, while f(x) ι + ε fails for some x arbitrarily large, no matter how small ε 0 is taken. That σ and ι must necessarily exist is a consequence of the completeness property of the real number system. Define a def = lim inf x→+∞ ψ(x) x and A def = lim sup x→+∞ ψ(x) x . Then the ε-x0(ε) bounds for ψ(x)/x can be reformulated as the statement that log(2) ≤ a ≤ A ≤ 2 log(2). The definitions of ψ and ϑ yield ψ(x) = pk≤x log(p) = ∞ k=1 p≤x1/k log(p) = ϑ(x) + ϑ(x1/2) + ϑ(x1/3) + · · · , and so ψ(x) − 2ψ(x1/2) = ϑ(x) − ϑ(x1/2) + ϑ(x1/3) − · · · .

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