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I was reading Riemann's paper, On the number of primes less than a given quantity, and found the following pharagraph.

The known approximating expression $\pi(x) = Li(x)$ is therefore valid up to quantities of the order $x^{1\over2}$ and gives somewhat too large a value

Since I'm not native English speaker, It was a little bit difficult for me to understand what does the phrase "up to" imply. I first guessed that the pharagraph above means $$\pi(x)=Li(x)+O(x^{1\over2})$$ but then realized it can't be true beacuse the error term is too small.(Since, as long as I know, the error term of $\pi(x)$ and Li(x) must be grater than $O(\sqrt x \log x)$). Can someone tell me the exact meaning of the pharagraph above?

비선형편미분방정식
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    Indeed, assuming RH we have $\pi(x)=Li(x)+O(\sqrt{x}\log(x))$. – Dietrich Burde Aug 16 '22 at 07:57
  • yeah, beacuse of that, the error term of $x^{1\over2}$ is impossible. So I sill don't know what does the phrase "up to quantities of the order $x^{1\over2}$" mean. @DietrichBurde – 비선형편미분방정식 Aug 16 '22 at 08:03
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    It's interesting Riemann calls a $O(\sqrt{x}\log x)$ difference "up to $x^{1/2}$", presumably because it's $o(x^q)$ for any $q>\frac12$. It's confusing by modern standards, but he's not a modern author. – J.G. Aug 16 '22 at 08:03
  • It is now known that $Li(x)>\pi(x)$ does not hold in general , in fact $Li(x)-\pi(x)$ switsches its sign infinite many often, but the first switch is at an extremely large value $x$. – Peter Aug 16 '22 at 08:53
  • Oh, that would be right. But, if so, then did Riemann stated the phrase "up to" on the assumption of Riemann hypothesis is true? @J.G. – 비선형편미분방정식 Aug 16 '22 at 10:01
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    I don't think Riemann assumed the RH in this analysis. – J.G. Aug 16 '22 at 10:22

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It means that the best approximate without using RH is order $x^{\frac12}$.

JMP
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