A different way to calculate number of primes less than a particular number x

Question

According to Wikipedia, series representation of logarithmic integral function is as follows:

Now, as per my findings from this we can also calculate number of primes less than a particular number x as follows: $$f(x) = \sum_{i=2}^{x^{1/\phi}} {i^{1/\phi}\over \ln (i)}$$ Where $\phi$ is the golden ratio.

This formula works at par with the logarithmic integral li(x) and as per my findings it works as prominent as logarithmic integral.

In below graphs, it can be seen that li(x) superimposes the above proposed function. (Both graphs have >1B Prime Numbers)

I am trying following things:

1. I am trying an alternative solution for estimating number of primes less than a particular number x.
2. I am trying to establish a relationship between golden ratio and prime numbers as it has always been speculated.
3. I am trying to establish a finiteness in estimating the number of primes as the method proposed is a finite series.

Is this work interesting and worth publishing? Shall I go forward with these findings?

Answer 1

The answer is `No' just because the statement is wrong. Take $x=2^\phi$. The rhs of your relation gives $$\frac{2^{1/\phi}}{\ln(2)}\simeq 2.21422$$ whereas $$li(2^\phi)\simeq 2.22625$$