Bounding pi(n): Dusart's 1999 Extension of Rosser's Theorem

[CRGreathouse]

I was looking at a paper of Dusart (his 1999 extention of Rosser's Theorem), where the author gives (without proof, but using the results proved in the rest of the paper) bounds for the prime counting function pi(n).
The minimum was \frac{x}{\log(x)}\left(1+\frac{0.992}{\log(x)}\right) (for x >= 599) and the maximum was \frac{x}{\log(x)}\left(1+\frac{1.2762}{\log(x)}\right) (for x >= 2).

Does anyone know of a stronger result for the upper bound, perhaps with a stronger restriction? In practice the lower bound sticks much closer to pi(n) than the upper bound. In any case Dusart gives a reference for his method of derivation, but I haven't tracked it down yet.

Edit: Since the new version of this site, I can't read it with Firefox -- the site crashes my browser.

 

[shmoe]

I've seen those Dusart bounds mentioned more than once, so I'm thinking they might be the best explicit bounds known but I could easily be wrong. Rosser's papers would be a place to look. You might try emailing Dusart. Do report back if you find something stronger!

 

[CRGreathouse]

Well, I did email Dusart, but I've since found an unpublished (as far as I can tell) paper of his with stronger bounds of a different form, valid for larger k.

If k >= 32299 then \pi(k)>\frac{k}{\ln k}\left(1+\frac{1}{\ln k}+\frac{1.8}{(\ln k)^2}\right).
If k >= 355991 then \pi(k)<\frac{k}{\ln k}\left(1+\frac{1}{\ln k}+\frac{2.51}{(\ln k)^2}\right).

These are actually phenomenal bounds.

 

[shmoe]

Interesting, thanks. Are those proved with a similar method? Do you have the reference handy?

 

[CRGreathouse]

Interesting, thanks. Are those proved with a similar method? Do you have the reference handy?

I'm actually not sure how to cite it. The paper is just a research report of sorts. I found it on a Columbia professor's page -- I'm not sure where it's actually from. I think it's building on the published paper, since it uses the main result of that paper, but it has one bound strictly worse than one in the paper. It's a little confusing.

Pierre Dusart, "Sharper bounds for psi, theta, pi, and p_k", http://www.math.columbia.edu/~glass/spring04/chebyshev.pdf , June 1998.

 

[shmoe]

That's great, thanks. I don't usually care about explicit bounds, but it's nice to know where they live.

 

[CRGreathouse]

That's great, thanks. I don't usually care about explicit bounds, but it's nice to know where they live.

Yeah, but these are killer bounds. I love the fact that both bounds are fairly tight and of the same form -- it made me look at

(\ln k)^2\left(\frac{\pi(k)\ln k}{k}-1-\frac{1}{\ln k}\right)

for large-ish k to see if it would be reasonable to use a narrower range yet for the constant (now 1.8 to 2.51). Empyrically, it seems that 2 could easily be a lower bound (perhaps even 2.15 or 2.2) for all large k, but that's just conjecture until we know more of the Riemann zeta zeros.

(I should really look at the form Dusart is using to prove this and see what I can get taking the limit as A -> infinity, the result that could be proved for large enough k assuming the truth of the RH.)

 

[shmoe]

At the time that article was written, it was known that the first 1.5 billion were on the critical line. Computations have gone *much* furthur since then, van de Lune hit 10 billion in 2001. The ZetaGrid project claims over 1134 billion and something like a billion per day, though I'm not sure what the most recent published result is.

At the rate it's increasing some kind of explicit bound as a function of the number of zeros might be interesting.

 

[CRGreathouse]

After studying the data, and noticing the reduction in variability as pi(k) is large, I conjecture the following:

(Edit: well, so much for that conjecture!)

Now all I have to do is look up Dasart's references and figure out how he does his calculations, then apply it with the known Riemann zeros and check the small values myself. (I presume the method gives a high effective lower bound which must be reduced manually.)

 

[CRGreathouse]

You might try emailing Dusart. Do report back if you find something stronger!

I've gotten in touch with Dusart. He sent me his entire thesis... 174 pages long... in French. Wish me luck, I'm diving in!