Can insight be derived from direct formulae for prime number functions?

Question

Dear StackExchange Community,

I am an amateur enthusiast and was attempting to construct a formula for the n th prime using elementary functions - I didn't achieve this* but I did come up with some formulae in finite products and sums of elementary functions for various number theoretic functions such as the Euler Totient, Prime Pi, Möbius Mu, Divisor Sigma, Next Prime, LCM, etc.

For example it can be easily verified from the property of cosine that the below expression is equal to the divisor sigma function σ_k(n) for each k,n. $$ σ_k(n)=\sum _{m=1}^n m^{k-1} \sum _{l=1}^m \cos \left(2 \pi \frac{l n}{m}\right)$$

*Edit: Below is an expression for the n'th prime as a function of n in elementary functions.

$$p(n)=\sum _{a=2}^{2^n} \sin \left(\pi 2{}^{\wedge}\left(\left(n-\sum _{b=2}^a \frac{\sin ^2\left(\frac{\pi }{b}((b-1)!)^2\right)}{\sin ^2\left(\frac{\pi }{b}\right)}\right){}^2-1\right)\right)\frac{a \sin ^2\left(\frac{\pi }{a} ((a-1)!)^2\right) }{\sin ^2\left(\frac{\pi }{a}\right)}$$

Given these methods are unnatural and elementary I suspect these results are not insightful or original, but what motivates me to comment is that none of the commonly cited examples of prime generating functions involve only elementary functions - they will employ congruence modulus or the least integer function or integral part or something related, and so I wonder then if the idea above is in fact novel in some way. The Prime Formulas article (Mathworld) gives the following account : "There exist a variety of formulas for either producing the nth prime as a function of n or taking on only prime values. However, all such formulas require either extremely accurate knowledge of some unknown constant, or effectively require knowledge of the primes ahead of time in order to use the formula (Dudley 1969; Ribenboim 1996, p. 186)." But in the above construction of Next Prime we can enumerate NP(n) (or just the inner sum enumerating j) to give a sequence that consists only of primes and is not constrained by the above criteria. So I am interested to know if it could be assessed whether such ideas could be fruitful or could benefit from some more sophisticated techniques to derive some insight about the distribution of prime numbers and related inquiries.

Thanks and regards,

Arthur E.

Answer 1

Every "elementary" formula I know of is a disguised implementation of a slow algorithm for testing whether a number is prime. For actually computing primes, it's better just to directly implement a fast primality testing algorithm, of which there are many. For actually proving something about primes, experience has shown that it's better to either ask for asymptotic rather than exact information or to use more sophisticated techniques (e.g. the Riemann zeta function).

A basic reason formulas like the one you give are not useful for proving anything is that they involve the cancellation of many terms, and there's no way to extract reliable asymptotic information without knowing much more about how the terms cancel.

Answer 2

C P Willans, On formulae for the $n$th prime, Math Gazette 48 (1964) 413-415 gives $$\pi(m)=\sum_2^m(\sin^2\pi{((j-1)!)^2\over j})/\sin^2(\pi/j))$$ This is quoted in Paulo Ribenboim, The Little Book of Bigger Primes.