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

In summary, the conversation discusses Dusart's proof that ##\theta (x)<x## for very large ##x##, where ##\theta (x)## is the first Chebyshev function. The best bounds currently known for ##\theta (x)## are ##|\theta(x) -x| < 0.007 \frac{x}{ln x}## for ##x>10^7##, but there is no proof for the tighter bound mentioned. The conversation also mentions searching the internet for information on the Chebyshev function and how Dusart's result relates to the asymptotic behavior of ##\theta(x)##.
  • #1
Physicist97
31
4
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)## ?
 
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  • #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?
 
  • #4
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##.
 
  • #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 :) .
 

1. What is the Chebyshev function ##\theta (x)## and what does it represent?

The Chebyshev function, denoted as ##\theta (x)##, is an important function in number theory that counts the number of prime numbers less than or equal to a given number x. It is also known as the first Chebyshev function or the prime-counting function.

2. How is the Chebyshev function related to the prime number theorem?

The prime number theorem states that the number of primes less than or equal to x is approximately ##\frac{x}{\ln x}##. The Chebyshev function is closely related to this theorem as it provides a more precise estimate of the number of primes by taking into account higher powers of x in the logarithmic term. In fact, the prime number theorem can be derived from the asymptotic behavior of the Chebyshev function.

3. What are the bounds on the Chebyshev function?

The Chebyshev function has both lower and upper bounds. The lower bound is given by ##\frac{x}{\ln x}## and the upper bound is given by ##1.25506\frac{x}{\ln x}##. These bounds are known as the Brun-Titchmarsh theorem and they provide a range within which the Chebyshev function must lie for all values of x.

4. How are the bounds on the Chebyshev function useful in number theory?

The bounds on the Chebyshev function are useful in proving various results in number theory. For example, they can be used to prove the existence of infinitely many twin primes (pairs of primes that differ by 2). They are also used in estimating the error term in the prime number theorem, which has important implications in the distribution of prime numbers.

5. Can the bounds on the Chebyshev function be improved?

Yes, the current bounds on the Chebyshev function are not the best possible. There have been several attempts to improve these bounds, with the current record being ##0.92129\frac{x}{\ln x}## for the lower bound and ##1.00019\frac{x}{\ln x}## for the upper bound. However, proving tighter bounds is a difficult problem and it remains an active area of research in number theory.

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