I Bounds on Chebyshev Function ##\theta (x)##

[Physicist97]

Hello,

I remember reading somewhere that Dusart proved that $\theta (x)<x$ for very large $x$. Where $\theta (x)$ is the first Chebyshev function (the sum of the logarithms of all primes less than or equal to $x$). I couldn't find any source for this and was wondering if anybody had one, or maybe knew Dusart's proof of it. Also I wondered what are currently the best bounds on $\theta (x)$ ?

 

[fresh_42]

http://www.unilim.fr/laco/rapports/1998/R1998_06.pdf

looks like $|\theta(x) -x| < 0.007 \frac{x}{ln x}$ for $x>10^7$+ something.

 

[Physicist97]

It seems no one has proven $\theta (x)<x$ . Is it simply hard to prove or has there been some counter-example to this bound?

 

[fresh_42]

Have you searched the internet on the Chebyshev function? There must be plenty of entries I guess.

Edit: Sorry, too late now for me to think about how Dusart's result match with the asymptotic behavior $\theta(x) \sim x$.

 

[Physicist97]

I have searched the Internet for quite some time now, but the best bounds I could find were the ones you linked to that Dusart published. I had heard mention on a similar sight that Dusart had proven the tighter bound I mentioned before, but no reference was given. That's why I thought of asking on here, and to me it seems that bound hasn't been proven yet. Thank you for the help, knowing $\theta (x)<x$ isn't proven yet is good enough info for me :) .