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

  1. Aug 20, 2016 #1
    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)## ?
     
  2. jcsd
  3. Aug 20, 2016 #2

    fresh_42

    Staff: Mentor

  4. Aug 20, 2016 #3
    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?
     
  5. Aug 20, 2016 #4

    fresh_42

    Staff: Mentor

    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##.
     
  6. Aug 21, 2016 #5
    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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