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One of my close friends and I have been working towards an exact prime counting function. The approach we have came up accurately produces the number of composite numbers that occur before a given real number What we are really interested in is (note that we are amateur mathematican) are the implications an exact prime counting function would bring on the mathematical community. In order to avoid ambiguity because my question is very specific towards the study of primes i have linked and noted a previous stack exchange post which is synonomous to my inquiry and which address almost all of my concerns but unfortunately the post is too old for any reasonable comment and reply from the original commentators of the post. Our approach to $\pi (x)$ consists of a collection of triple and quadruple summations being added together An example from our work is $$\sum_{n_{2}=0}^{3} \sum_{n_{1}=0}^{4} \sum_{n=0}^{\frac{q-\left(33+44n_{3}\right) +\left(6n_{2}+8n_{3}\right)}{110+20n_{2}}} \frac{q-\left(10n+20n_{2}\right) + \left(3+4n_{3} \right) + \left(6n_{2}+8+n_{3} \right)} {100n+30+40n_{3}}$$

This plus 5 other quadruple and triple summations are being added to each other and are included in our approach. And please notice that in the post I posted below the author does use an example of double summations but is rendered useless for it is not exact and very difficult to extract large values from and further explanation on why that is would be appreciated.

My first question is what are the impacts an exact (that means not just useful and a really good estimate but able to compute all values of $\pi (x)$ to their actual values) prime counting function would have the mathematical community or if impacts seems too broad of a term to use then of what importance is it to the world of open sourced math problems and math hobbyists such as myself? Would this be considered a elementary function as opposed to the approaches made made by Dirichlet and Riemann using complex analysis in which I would like to note are extremely useful but not exact prime counting functions?

Though I find the comments to be somewhat ambiguous on known formulas for exact prime counting functions I would appreciate clarification on them as well as an extended response to whether there is an "exact" $\pi (x) $. If the stack exchange community could also direct us to whom or what type of academic body we should contact in order to better present our work and be able to better directly question those who could provide insight on this subject matter then that is the last of our concerns.

Thank who ever is able to answer any of our questions and we encourage any one needing to edit this post so as to fix our latex or in order to provide clarity to others reading this post please feel free to do so.

The myth of no prime formula? And these two proved to be useful https://en.m.wikipedia.org/wiki/Prime-counting_function

http://mathworld.wolfram.com/PrimeCountingFunction.html

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Euler77
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  • Note that (unless there is a typo) your formula involves division by zero. – David Sep 23 '15 at 01:33
  • If the question is really "what would the impact be ...?" then the rest of your question is just noise. What is your actual question? The key factor about such a formula would be performance. $$\sum_{k=1}^n \left((k-1)(k-1)!\bmod k\right)$$ is a formula which works. – Thomas Andrews Sep 23 '15 at 01:40
  • The above sum is a bit odd - is the upper limit of the outer sum really $$\frac{q-3}{q-5q}?$$ Because $q-5q=-4q<0$ unless $q<0$. – Thomas Andrews Sep 23 '15 at 01:48
  • @Thomas Andrews I'm afraid I have made a mistake on the upper bound of on one of summations I displayed inorder to give an example of our approach to this problem and i have changed it to a correct example from our paper but unfortunately my novice latex skills prevent me from doing it so. – Euler77 Sep 23 '15 at 14:32
  • @Thomas Andrews I would like to bring THREE specific questions that posed in my inquiry to your attention The first you understood but gave no response to which was what are the implications of an equation like this – Euler77 Sep 23 '15 at 14:33
  • Second is would the example (which I have failed to accurately provide to you) be considered an "elementary" function? My Third question is to whom should we properly present our work to? – Euler77 Sep 23 '15 at 14:39

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