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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)## ?