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Various bounds and computations for Chebyshev's functions $$ \vartheta(x) = \sum_{p\le x} \log p, \quad \psi(x) = \sum_{p^a\le x} \log p $$ can be found in e.g.

Nazardonyavi and Yakubovich cite Ingham to give as Theorem 1.14 $$ \psi(x)-x = \Omega_{\pm}\left(x^{1/2}\log\log\log x\right) $$ and Wikipedia cites Hardy and Littlewood already with $$ \psi(x)-x \neq o\left(x^{1/2}\log\log\log x\right) $$ (though it doesn't say explicitly that large deviations should be on both sides, but follows the two-sided result of Schmidt). These suggest that there should be infinitely many $x$ with $\psi(x)-x>A\sqrt{x}$ for any given constant $A$.

But Dusart gives for all $x>0$ $$ \psi(x)-\vartheta(x)<1.00007\sqrt{x}+1.78\sqrt[3]{x} $$ which is bounded by a constant multiple of $\sqrt{x}$.

Hence (if I read correctly) there should be infinitely many $x$ with $\vartheta(x)>x$. Is any such value for $x$ known, or a range where this is expected to occur?

Glorfindel
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Zander
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    Some more estimates on $\theta(x)$ are given here; also $\theta(x)>0.99871149x$ for all $x\ge 10^8$, which is of course not what you have asked, but interesting. – Dietrich Burde Apr 10 '15 at 08:25
  • Do the numbers represented in decimal notation represent exact rational numbers or floating point approximations to numbers of unknown algebraicity? – Adam Ledger May 01 '20 at 05:50

2 Answers2

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Littlewood’s oscillation theorem says that there is a positive constant $C$ such that for infinitely many numbers $x$, $$ \theta(x) > x + C\sqrt{x}\log \log \log x. $$ A discussion about this can be found in the article Efficient prime counting and the Chebyshev primes. It is related to the study of $\epsilon(x)=\mathrm{Li}(x)-\pi(x)$, and $\epsilon_{\theta(x)}=\mathrm{Li}(\theta(x))-\pi(x)$. The function $\epsilon(x)$ is known to be positive up to the very large Skewes’ number. On the other hand it is known that RH is equivalent to the fact that $\epsilon_{\theta(x)}$ is always positive. Concerning $\theta$ see also Lemma $1.6$ in the above article for $\theta(p_{n+1})>p_{n+1}$.

Aaron Meyerowitz
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Dietrich Burde
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The analogous problem of finding an explicit number $x$ where $\pi(x) > \mathop{\rm li}(x)$ is related to "Skewes' number". For that problem, Bays and Hudson show that such an $x$ can be found less than about $1.4\times10^{316}$; this has been modestly improved a few times since then (I think most recently by Zegowitz). Based on approximate computations, it seems reasonable to suspect that that's actually the size of the smallest such $x$.

People don't seem to have looked as much at specific $x$ for which $\theta(x)>x$, but the same techniques apply, and I suspect that the smallest such occurrence is basically the same as that of $\pi(x) > \mathop{\rm li}(x)$.

Greg Martin
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