We restrict primes and prime powers to sets H(x) = ∪_(n=1)~∞(x/2n, x/2n-1]. Let θ_H(x) = Σ from p∈H(x) of log p. Then the error in θ_H(x) has, unconditionally, the expected order of magnitude θ_H(x) = x log 2 + O(x~(1/2)). However, if ψ_H(x) = Σ from p~m∈H(x) of log p then ψ_H(x) = x log 2 + O(log x). Some reasons for and consequences of these sharp results are explored. A proof is given of the "harmonic prime number theorem", π_H(x)/π(x) → log 2.
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