Abstract
In this paper, we study generalised prime systems for which both the prime and integer counting functions are asymptotically well-behaved, in the sense that they are approximately li ( x ) and ρ x , respectively (where ρ is a positive constant), with error terms of order O ( x θ 1 ) and O ( x θ 2 ) for some θ 1 , θ 2 < 1 . We show that it is impossible to have both θ 1 and θ 2 less than 1 2 .
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