On the first sign change of θ(x)−xθ(x) - x

preprint2014arXivOpen access

Abstract

Let θ(x)=∑p≤xlog⁡pθ(x) = \sum_{p\leq x} \log p. We show that θ(x)<xθ(x)<x for 2<x<1.39⋅10172<x< 1.39\cdot 10^{17}. We also show that there is an x<exp⁡(727.951332668)x<\exp(727.951332668) for which θ(x)>x.θ(x) >x.

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