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Published online by Cambridge University Press: 17 April 2009
We restrict primes and prime powers to sets . Let
. Then the error in θH(x) has, unconditionally, the expected order of magnitude
. However, if
then ψH (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”, πH (x)/π (x) → log 2.