A prime counting function I have never seen

[logistic_guy]

It gets more accurate the higher you go.

I am an engineer, so I understand things better when I see some calculations!

 

[binbots]

I am an engineer, so I understand things better when I see some calculations!

 

[logistic_guy]

That was a beautiful start. I have seen before the prime number theorem which is $\displaystyle \sim \frac{x}{\ln x}$ as well as Ramanujan's formula $\displaystyle \frac{x}{\ln x - 1}$.

Also the $\displaystyle \text{Li}(x)$ function is very famous in the integral world.

$\displaystyle \text{Li}\left(10^{300}\right) = \int_{2}^{10^{300}} \frac{1}{\ln x} \ dx \approx 1.449750053 \times 10^{297}$

So, the main idea of this thread is that your discovery is closer to the $\displaystyle \text{Li}(x)$ function than other approximation methods?

It gets more accurate the higher you go.

Now I understand what you mean here because I know that the logarithmic integral $\displaystyle [ \ \text{Li}(x) \ ]$ yields better approximation for the number of prime numbers as $\displaystyle x$ gets bigger.

Let me apply your formula to my example.

$\displaystyle \large \pi(100) = \frac{100}{\ln 100 - e^{\frac{1}{\ln 100}}} \approx 30$

It is still not a bad approximation compared to the actual value which is $\displaystyle 25$ while we used here a very small number.

 

[binbots]

That was a beautiful start. I have seen before the prime number theorem which is $\displaystyle \sim \frac{x}{\ln x}$ as well as Ramanujan's formula $\displaystyle \frac{x}{\ln x - 1}$.

Also the $\displaystyle \text{Li}(x)$ function is very famous in the integral world.

$\displaystyle \text{Li}\left(10^{300}\right) = \int_{2}^{10^{300}} \frac{1}{\ln x} \ dx \approx 1.449750053 \times 10^{297}$

So, the main idea of this thread is that your discovery is closer to the $\displaystyle \text{Li}(x)$ function than other approximation methods?


Now I understand what you mean here because I know that the logarithmic integral $\displaystyle [ \ \text{Li}(x) \ ]$ yields better approximation for the number of prime numbers as $\displaystyle x$ gets bigger.

Let me apply your formula to my example.

$\displaystyle \large \pi(100) = \frac{100}{\ln 100 - e^{\frac{1}{\ln 100}}} \approx 30$

It is still not a bad approximation compared to the actual value which is $\displaystyle 25$ while we used here a very small number.

Yes the goal here is to find the best approximation only using arithmetic in order to get a more intuitive understanding of the primes.