# I Pi(x) from zeta

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1. Jan 8, 2018

In the last part of https://en.wikipedia.org/wiki/Riemann_zeta_function#Mellin-type_integrals, I read two expressions of Riemann's zeta function ζ(s) in terms of s and of integrals of the prime-counting function π(x) (the second one using Riemann's prime-counting function J(x) from which, the article states, the usual prime counting function π(x) can be recovered) . However, since there is no way to determine π(x) for an arbitrary x (only good approximations), but there are formulas to calculate ζ(s). I conclude that there is no way to solve for π(x) (or J(x)) from these expressions. Is this correct?
[There seems to be no rubric for number theory in the forums, so I am posting this here.]

2. Jan 8, 2018

### Staff: Mentor

3. Jan 8, 2018

Thank you, fresh_42. I have just downloaded the zip file you referred to. Also, very interesting tip about switching languages: I took a look at the other five languages that I can easily read, and at a quick glance confirmed that the content differs from language to language. Did you have a particular language in mind?

4. Jan 8, 2018

### Staff: Mentor

Where did you get a zip file from? I just clicked on the pdf download and that's it. It directly opened in the browser (76 pages).

5. Jan 8, 2018

### Staff: Mentor

There is no known and efficient way. It can be done in O(x2/3/(log(x))2), and we have a value for 1026. Source.

If there would be an efficient way to determine the prime counting function exactly, we could use this as primality test, so obviously it has to be at least as complicated as the easiest primality test.

There are various formulas linking the different things. wolfram.com has a collection.

6. Jan 8, 2018

Thanks again, fresh 42. I don't know why I got a zip file the first time; upon receiving your post, I tried again and was able to download the pdf file properly. Super!
I see that the whimsical fictional element "Riemannium" ended up being taken very seriously:
https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.118.130201

7. Jan 14, 2018

### aheight

That is not true. Riemann, in his 1859 paper on the prime counting function introduces an "exact" formula for $\pi(x)$. But it's not easy. His formula can be more simply derived using the Residue Theorem.

Last edited: Jan 14, 2018
8. Jan 14, 2018

Thanks, aheight. I presume you are referring to the following?:
J(x) = li(x)-∑ρ li(xρ)-ln2+∫x dt/[t*(t2-1)*ln(t)] =∑(n=1 to ∞)π0(x1/n/n) where
ρ ranges over all non-trivial zeros of the Riemann zeta function, and
π0(x) = ½ limh→0(π(x+h)-π(x-h))

9. Jan 14, 2018

### aheight

Yes.

10. Jan 17, 2018