Broughan, K.A. & Casey, R. J. (2005). Harmonic sets and the harmonic prime number theorem. Bulletin of the Australian Mathematical Society, 71, 127-137.
Permanent Research Commons link: http://hdl.handle.net/10289/2038
We restrict primes and prime powers to sets H(x)= U∞n=1 (x/2n, x/(2n-1)). Let θH(x)= ∑ pεH(x)log p. Then the error in θH(x) has, unconditionally, the expected order of magnitude θH (x)= xlog2 + O(√x). However, if ψH(x)= ∑pmε H(x) log p then ψH(x)= xlog2+ 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) → log2.
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This article has been published in the journal: Bulletin of the Australian Mathematical Society. ©2005 Australian Mathematical Society. Used with Permission.