Logarithmic Integral and Primes

In summary, The conversation discusses the use of two integrals - ∫x/log(x) dx and ∫1/log(x) dx - for approximating π(x), with the latter being the actual definition of li(x). While the first integral is commonly used and closely matches π(x), the second integral is a better approximation with more accurate error estimates. Mathematica only recognizes the second integral as the logarithmic integral.
  • #1
Frogeyedpeas
80
0
Hey guys, I was reading a brief article which described the logarithmic integral for approximating π(x)

in two ways:

∫x/log(x) dx

and

∫1/log(x) dx

I am aware that the second is the actual definition of li(x) but the top is used extremely frequently and upon trying out the top it matches pi(x) very closely so I'm not sure which is correct or if both are in the running for being defined as logarithmic integral (though mathematica says only the second is)
 
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  • #2
Anybody?
 
  • #3
I don't understand what statement you are making about the top integral, but If you remove the integral sign and dx, you have an expression that according to the PNT grows like π(x).

Your second integral, taken from 2 to x, also known as li(x), is an even better approximation to π(x)
 

What is the logarithmic integral function?

The logarithmic integral function, denoted as Li(x), is a special function in mathematics that is defined as the integral of the reciprocal of the natural logarithm of x. In other words, it is the antiderivative of 1/ln(x). It is closely related to the prime counting function and is used in number theory to estimate the distribution of prime numbers.

What is the connection between the logarithmic integral function and prime numbers?

The connection between the logarithmic integral function and prime numbers is through the Prime Number Theorem, which states that the number of primes less than or equal to x is approximately equal to Li(x). In other words, Li(x) can be used to estimate the number of prime numbers up to a certain value x. This is a very important result in number theory and has many applications.

How is the logarithmic integral function used in the Riemann hypothesis?

The Riemann hypothesis is one of the most famous unsolved problems in mathematics and is closely related to the distribution of prime numbers. The logarithmic integral function is used in the proof of the Riemann hypothesis, as it helps in understanding the asymptotic behavior of the prime counting function. However, the Riemann hypothesis still remains unproven.

What are some other applications of the logarithmic integral function?

The logarithmic integral function has many applications in mathematics and physics. It is used in complex analysis, number theory, and statistical mechanics. It also has applications in the study of black holes and the distribution of energy levels in quantum systems. Additionally, it is used in the analysis of the error term in the Prime Number Theorem and in the study of the zeta function.

Can the values of the logarithmic integral function be calculated exactly?

No, the values of the logarithmic integral function cannot be calculated exactly for all values of x. This is because it is an improper integral and does not have a closed form solution. However, it can be approximated numerically using various methods, such as the Euler-Maclaurin formula or numerical integration techniques. It is also possible to calculate the values for certain special cases, such as when x is a prime number.

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