# Is this formula for the $n^{th}$ prime number useful?

Is the below formula for the $n^{th}$ prime number in elementary functions useful somehow?

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

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How widely have you looked for elementary formulas for primes? Have you looked at Hardy and Wright? at Crandall and Pomerance? at Niven, Zuckerman, and Montgomery? at MathSciNet? – Gerry Myerson Jun 9 '11 at 12:30
Hello Gerry, please forgive my ignorance but I am not familiar with these authors. – Athanasios Evangelou Jun 9 '11 at 13:46
@A. E.: if you're not familiar with G.H. Hardy, then you have some rather delightful reading ahead of you! And I heartily encourage you to read more into the number theory; believe it or not, things get even more fascinating once you get past the 'surface' of quirky formulas like the one you found and into the meat of the subject. – Steven Stadnicki Jun 10 '11 at 2:14
Eh, I do not know if this is the place to say so, since the lack of experience, but, judging from my narrow view, the branch of mathematics named analytic number-theory is $not$ number-theory at all; moreover, as Weil put it, Hardy is not a number-theorist. If one can gain insights into number-theory, as the author wants, then tell me so as to correct my wrong view, thanks. P.S. My view does not matter as a commonly accepted knowledge, and I put it here just to judge this. – awllower Jun 10 '11 at 7:37
"analytic number theory is not number theory" is meaningless without a definition of terms. – Gerry Myerson Jun 10 '11 at 12:24

I would argue that by your definition it is non-elementary in exactly one fashion, since $$n!=\prod _{i=1}^n i=\exp \left(\sum _{i=1}^n \ln (i)\right)$$, but I get the point ! – Athanasios Evangelou Jun 10 '11 at 2:33
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.