Expansion for the prime counting function

In summary, we are discussing possible expansions for the prime number \pi(x) and the logarithmic integral function Li(x). While there is an asymptotic expansion for Li(x), it is not convergent and other series for Li(x) are more complex. It is still a sought-after goal to find a nice analytic series for \pi(x).
  • #1
zetafunction
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my question is, let us suppose we can find an expansion for the prime number (either exact or approximate)

[tex] \pi (x) = \sum _{n=0}^{\infty}a_n log(x) [/tex]

and we have the expression for the logarithmic integral

[tex] Li (x) = \sum _{n=0}^{\infty}b_n log(x) [/tex]

where the numbers a(n) and b(n) are known , then my question is , what could one expect about the difference expansion

[tex] \pi (x) - Li(x) = \sum _{n=0}^{\infty}(a_n - b_n) log(x) [/tex] ??
 
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  • #2
Unless the a_i and b_i are functions of x, no such expansions exist as they would imply that pi(x)/log(x) and Li(x)/log(x) are constant.
 
  • #3
[tex] \pi (x) = \sum _{n=0}^{\infty}a_n log^{n} (x) [/tex]

[tex] Li (x) = \sum _{n=0}^{\infty}b_n log^{n} (x) [/tex]

sorry i made a mistake it should include powers of log(x) and not only logarithm of x , sorry about that.

[tex] \pi(x) - Li (x) = \sum _{n=0}^{\infty}(a_n - b_n) log^{n} (x) [/tex]
 
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  • #4
About what point are you taking this expansion? I don't see a way to take it about the point at infinity.
 
  • #5
"the expression for the logarithmic integral"

While there is an asymptotic expansion for the logarithmic integral of the form you propose it is not convergent. It is only a good representation of li(x) for a certain number of terms depending on x.

Other series for li(x) that do converge are much trickier. For example Ramanujan's series: http://en.wikipedia.org/wiki/Logarithmic_integral_function#Series_representation.

Pi(x) - Li(x) = (Difference of Pi(x) - Li(x)) (see wiki on RH for the best estimate on this) + (Error on your expression for Pi(x)) + (Error on your expression for Li(x))

Note that I added the errors because the sign of the errors is not known. If you could prove the errors are strictly positive you could subtract them.

Of course everyone wishes that such a nice analytic series exists for Pi(x). Many eminent mathematicians have been looking for such a series for hundreds of years with not much to show for it.
 
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What is the prime counting function?

The prime counting function is a mathematical function that gives the number of prime numbers less than or equal to a given number. For example, the prime counting function of 10 would be 4, since there are 4 prime numbers (2, 3, 5, and 7) less than or equal to 10.

Why is the prime counting function important?

The prime counting function is important because it helps us understand the distribution of prime numbers and their relationship to other numbers. It also has applications in number theory, cryptography, and other fields of mathematics.

What is an expansion for the prime counting function?

An expansion for the prime counting function is a series of terms that approximates the value of the function for larger numbers. This expansion is useful because it allows us to estimate the number of prime numbers without having to manually count them.

How accurate is the expansion for the prime counting function?

The accuracy of the expansion for the prime counting function depends on the number of terms used in the series. Generally, the more terms that are included, the more accurate the approximation will be. However, for extremely large numbers, the accuracy may still be limited.

Are there other methods for approximating the prime counting function?

Yes, there are other methods for approximating the prime counting function, such as the Riemann zeta function and the Möbius function. These methods may have different levels of accuracy and applicability depending on the specific situation.

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