# Prime Counting Function

1. Apr 1, 2014

### xiang

Dear fellow learners,

Through an extracurricular project I have found a really cool equation to count primes. The equation can evaluate
Pi(x)+Pi(√x)/2+Pi(cubedroot(x))/3+...Pi(nthroot(x))/n

I have directly proved my equation so I now it will be accurate 100% of the time. Although the equation is cumbersome my prof thinks it is a really cool idea and that I should either try and use it to prove some prime conjectures or simply publish it. However, he is not a specialist in number theory and suggests I consult several forums to find any equations that resemble mine.

So, my question: are there any equations that can calculate
Pi(x)+Pi(√x)/2+Pi(cubedroot(x))/3+...Pi(nthroot(x))/n?

2. Apr 1, 2014

### Deedlit

This function is known as the Riemann Prime Counting function. An explicit formula was proposed by Riemann and later proven by Mangoldt; it involves the nontrivial zeros of the Riemann zeta function and an improper integral. More information can be found at the Mathworld web site.

I don't know how good your formula is for computation, but the best known methods for computing pi(x) are the Lagarias-Odlyzko-Miller method, which computes pi(x) in time O(x^(2/3)), and the analytic Lagarias-Odlyzko method, which has time O(x^(1/2)). Pi(x) has been computed up to 10^25.

3. Apr 1, 2014

### xiang

Thanks deedlit for the response!

I did did some research into the Riemann prime counting function and while his searches for the same thing it requires a möbius function where mine does not. Is this difference enough to publish?

4. Apr 2, 2014

### Deedlit

The expression for the Riemann prime counting function does not use the moebius function; the moebius function is used to convert for the Riemann prime counting function to the regular prime counting function. You said that your equation was for the Riemann prime counting function, so to convert it the the regular prime counting function you would naturally use the moebius function as well, I would think.

Is your expression significantly different from the one given on the Mathworld web page? It would help to know exactly what your equation is. If you are reluctant to divulge it, I suppose you could write it up and upload it to the Arxiv site, so that you could have a record that you came up with it first, and then check with an expert if it is publishable. Others may chime in on whether this is a good idea or not.