On the first sign change of $\theta(x) - x$

Dave Platt; Tim Trudgian

On the first sign change of $\theta(x) - x$

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arxiv 1407.1914 v1 pith:34LLNM7W submitted 2014-07-08 math.NT

Dave Platt , Tim Trudgian This is my paper · ORCID

classification math.NT

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abstract

Let $\theta(x) = \sum_{p\leq x} \log p$. We show that $\theta(x)<x$ for $2<x< 1.39\cdot 10^{17}$. We also show that there is an $x<\exp(727.951332668)$ for which $\theta(x) >x.$

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Behaviour of the sequence $\vartheta_n = \vartheta(p_n)$
math.NT 2025-07 reject novelty 5.0 of 10

Replacing the n-th prime by the sum of logarithms of the first n primes makes analogues of Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures provable theorems.

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