I |Li(x) - pi(x)| goes to 0 under RH?

nomadreid

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Extremely quick question:
According to http://mathworld.wolfram.com/PrimeNumberTheorem.html, the Riemann Hypothesis is equivalent to
|Li(x)-π(x)|≤ c(√x)*ln(x) for some constant c.
Am I correct that then c goes to 0 as x goes to infinity?
Does any expression exist (yet) for c?
Thanks.
 

fresh_42

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Extremely quick question:
According to http://mathworld.wolfram.com/PrimeNumberTheorem.html, the Riemann Hypothesis is equivalent to
|Li(x)-π(x)|≤ c(√x)*ln(x) for some constant c.
Quick answer:
Am I correct that then c goes to 0 as x goes to infinity?
No. ##c## is a constant factor and goes nowhere. However, there are better approximations which may depend on x.
Does any expression exist (yet) for c?
Thanks.
Wikipedia has ##c=\dfrac{1}{8\pi}## found by L. Schoenfeld.
 

nomadreid

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Super! Thanks, fresh_42
 
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Wikipedia has ##c=\dfrac{1}{8\pi}## found by L. Schoenfeld.
For x>2657 (and only if the Riemann hypothesis is true, of course). Smaller values probably require a larger c, and we have to exclude x=1 of course.
 

nomadreid

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Thanks for the precision, mfb.
I followed up on fresh_42 's remark
there are better approximations which may depend on x.
to find better approximations, and found one at one source that I have not been able to find elsewhere: the first entry on
https://www.quora.com/What-is-the-relationship-between-the-Riemann-Hypothesis-and-prime-numbers
gives an exact formula which depends on the zeros of the zeta function under the assumption that the RH is correct and p primes: via a Möbius tranformation of:
formula.PNG


PS. After writing this, I noticed https://en.wikipedia.org/wiki/Prime-counting_function#Exact_form, which I presume is equivalent to the above.
 

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fresh_42

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For x>2657 (and only if the Riemann hypothesis is true, of course). Smaller values probably require a larger c, and we have to exclude x=1 of course.
As an add-on: This is standard for big-O notations, that they are only valid from a certain level on. E.g. the matrix exponent for 2 x 2 matrices is exactly ##\log_27## but there are better bounds for larger ##n##, but it doesn't make the constant a variable.
 

nomadreid

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After letting these answers ferment in my head for a while, I conclude that the answer to my implicit question about why we talk about approximations if there are exact formulas such as the one listed in the Wikipedia page
Counting.PNG

where μ(n) is the Möbius function, is that although this formula would be indeed exact (assuming RH), it is so tedious as to not be of much use even if we had an efficient method for producing non-trivial zeros, just as the brute-force counting is indeed an exact algorithm but of no use for large numbers, so that the approximations referred to are more important because they are reasonably tractable. That is, we sacrifice precision for speed. I hope this is an accurate representation?
 

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