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

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)## ?
 
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

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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##.
 
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 :) .
 

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