Is my equation for counting primes unique or similar to existing equations?

[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?

Thanks everyone for your time!

 

[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.

 

[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?

 

[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.

 

[CellCoree]

Dear fellow learner,

Thank you for sharing your equation for counting primes. It is always exciting to discover new approaches and ideas in mathematics. However, as a scientist, I must caution that it is important to thoroughly research and compare your findings with existing equations before claiming it as unique or groundbreaking.

In regards to your specific equation, I am not aware of any existing equations that directly calculate Pi(x)+Pi(√x)/2+Pi(cubedroot(x))/3+...Pi(nthroot(x))/n. However, it does bear some similarities to the Prime Counting Function, which is commonly used to estimate the number of primes below a given number. It also resembles the Riemann zeta function, which is used to study the distribution of prime numbers.

I would recommend consulting with experts in the field of number theory and conducting a thorough literature review to ensure that your equation is indeed unique and not simply a variation of existing equations. This will also help you to better understand the significance and potential applications of your discovery.

Best of luck in your future endeavors and keep exploring the fascinating world of mathematics!