A very good coincidence occurred today with me. While just plotting random functions in Mathematica, I entered this command:
Plot[(1/(8Pi))Sqrt[x]Log[x]-Abs[PrimePi[x]-LogIntegral[x]], {x, 1, 100}]
It gave the following output:
enter image description here
I increased the plotting limit (if it is called so) to 5000:
enter image description here
Now I increased it to 10000:
enter image description here
I increased the plotting limit even more, and it seems like that for all numbers bigger than a specific number (which is near 2000) the function I plotted is always greater than zero. I mentioned that a coincidence occurred. It was that I found the exact inequality on this article (see the "Consequences" section) and that's its connection with the Riemann hypothesis. The "specific number" that I mentioned was 2657. The inequality $\frac{1}{8\pi}\sqrt{x}\log(x)-|\pi(x)-\mathrm{li}(x)|>0$ (this is the function I plotted) holds for all $x\geq2657$ if the Riemann Hypothesis is true.
Now, I tried to generalize this. I plotted $\frac{1}{8\pi}\sqrt{x}\log^2(x)-|\pi(x)-\mathrm{li}(x)|$ from 1 to 5000. Here's the plot:
enter image description here
It goes greater than zero before 2657. So I decreased the limits to see the number. I plotted it from 1 to 50, and it seems like that the number $a$ such that $\frac{1}{8\pi}\sqrt{x}\log^2(x)-|\pi(x)-\mathrm{li}(x)|>0$ for all $x\geq a$ is 41. I didn't show the plot because I don't want the whole post to be filled with plots. I similarly did it for the function $\frac{1}{8\pi}\sqrt{x}\log^3(x)-|\pi(x)-\mathrm{li}(x)|$, and found out that the number such that for all $x$ greater than or equal to that number, this function is greater than zero is 13. I did it for $\frac{1}{8\pi}\sqrt{x}\log^4(x)-|\pi(x)-\mathrm{li}(x)|$, and found out that the number was 7. This gives the decreasing sequence: $$2657,41,13,7,5,5,...$$ The next number is not an integer, I can't figure out its value. I have four questions:

  • What is known about the sequence that I mentioned?
  • The inequality $\frac{1}{8\pi}\sqrt{x}\log(x)-|\pi(x)-\mathrm{li}(x)|>0$ is true for all $x\geq2657$. So are the functions that I mentioned greater than 0 for all $x\geq a$ ($a$ is one of the numbers of the sequence, it depends on the function that which $a$ we choose from the sequence) if the Riemann hypothesis is true?
  • What is the limit of this sequence? 0, negative infinity, or any other number?
  • The functions that I mentioned are not continuous, but if zoom out further, they look like continuous curves. So are there continuous functions $f_1(x),f_2(x),...$(one $f$ for each function I mentioned), such that $\lim_{x\to\infty}\frac{F_n(x)}{f_n{x}}=1$, where $F$ are the discontinuous functions that I mentioned, and $f$ are there corresponding continuous functions?($f$ is to $F$ what $\mathrm{li}(x)$ is to $\pi(x)$.). Any help would be appreciated.
    Note: This sequence is not in OEIS. Please give some more terms of the sequence. I think there is a way OEIS can recognize this. Since the next term is not an integer, it can be converted to a fraction with a denominator not equal to one. Now convert all the terms of the sequence to fractions in their simplest form. The numerators can't be less than 1, and I think that it's not sure that the numerators and denominators both form a decreasing sequence. Entering the numerators (or denominators) to OEIS, I think it can recognize it, if not, can we recognize it or give some information about it? Some hours later I will update this question to tell what I did.
    Update: This sequence is weird; I can't figure out the exact value of the next term, but its value is approximately 4.3265. The next term is approximately 3.86. If I didn't make a mistake in calculating, the next term is approximately 3.54628. Is this sequence converging to 3?
    Update:Emil Jeřábek's comment helped to deduce that the limit of this sequence is $e$. So one of the questions is solved. Another question, which is not that important, but I am still asking it:
  • What is the exact form of the 7th term of this sequence?
  • $\begingroup$ Please add some tags if you know one that is suitable for this question. $\endgroup$ – Leonhard Euler Oct 27 '20 at 12:09
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    $\begingroup$ Note that $\frac{1}{8\pi}\sqrt{x}\log^n(x)-|\pi(x)-\mathrm{li}(x)|>\frac{1}{8\pi}\sqrt{x}\log(x)-|\pi(x)-\mathrm{li}(x)|$ for all $n$. Hence for any $n$ existence of the value you ask for follows from RH, and this value is at most $2657$. On the other hand, existence of any of those values implies RH (this is much less trivial). $\endgroup$ – Wojowu Oct 27 '20 at 12:15
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    $\begingroup$ For large $n$, $\frac1{8\pi}\sqrt x(\log x)^n$ is approximately $0$ until a bit below $x=e$, when it starts to rapidly increae until a bit above $x=e$ when it becomes $\infty$ for all intents and purposes. Thus, the limit of your sequence is $e$. Moreover, since $|\pi(x)-\mathrm{li}(x)|$ is positive and smooth near $e$, the $n$th term in the sequence is roughly the $x$ such that $\frac1{8\pi}\sqrt x(\log x)^n\approx|\pi(e)-\mathrm{li}(e)|$, that is, it is $e+cn^{-1}+O(n^{-2})$ where $c=e\bigl(\log|1-\mathrm{li}(e)|-\frac12+\log(8\pi)\bigr)$ barring a numerical mistake. $\endgroup$ – Emil Jeřábek Oct 28 '20 at 8:37
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    $\begingroup$ @epic_math Looking at this again, this is actually the less trivial part (it would be harder without the absolute value bars; then you need something like Theorem 15.2 from Montgomery-Vaught). If RH fails and there is a zero of real part greater than some $\theta>1/2$, then the difference between $\pi(x)$ and $li(x)$ will infinitely often exceed $x^\theta$. For if this were not the case we could use this fact to get a convergent expression for $\zeta'/\zeta$ on the domain $Re(s)>\theta$, which is impossible in the presence of the zero. Comparatively, von Koch's result is much harder. $\endgroup$ – Wojowu Oct 28 '20 at 9:58
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    $\begingroup$ By the way, the few integer values in your sequence are of course due to the jump discontinuities of $\pi$ at primes, but your other “exact” values are numerical errors: e.g., the $n=8$ value is not “exactly 3.86”, but $\approx 3{.}85920$ (wolframalpha.com/input/…). Noninteger terms of the sequence are extremely unlikely to be rational (let alone terminating decimal), or even transcendental, or to have any reasonable closed-form expression. $\endgroup$ – Emil Jeřábek Oct 28 '20 at 15:18

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