Exercises 39 b) Find a constant c(θ) such that n≤x [x/n]−θ = c(θ)x + O(1) holds uniformly for θ ≥ 0. Uniformity means that the constants C and x0 that are implicit in the O-term do not depend on the parameter θ. (53) Show that n≤x log(φ(n)) = x log(x) + (c − 1)x + O(log(x)) where c = p 1 p log 1 − 1 p . Find the average order of log(φ(n)). Calculate the local average and choose the length h of the interval in terms of x so as to minimize the error term. (54) a) Use the identity n≤x x n − ψ x n − 2γ = D(x) − T(x) − 2γ[x] and the Dirichlet bound in the divisor problem to show that ψ(x) = O(x) independently of the Chebyshev method. This idea is due to Nina Spears. Her approach may be refined to yield close upper and lower numerical bounds for ψ(x)/x. Though unfortunately only at the cost of extensive computation. Note that this establishes the bound ϑ(x) = O(x) and Proposition 1.9 without reliance on the method of Chebyshev. b) Show that x/Kp≤x log(p) p = log(K) + O(1) where K ≥ 1 is an arbitrary constant, and the error is uniform in K. c) Apply b) to show that ϑ(x) x without reliance on the method of Chebyshev (H. N. Shapiro). (55) † Find an asymptotic estimate for the sum n≤x d(n) log(n) + 2γ , with a bound for the error term.

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