The n th Prime Exponentially

Consider both the Logarithmic integral, Li ( x ) = lim ϵ → 0 ∫ 0 1 − ϵ d u ln u + ∫ 1 + ϵ x d u ln u , and the prime counting function π ( x ) = ∑ p ≤ x 1 . From several recently developed known effective bounds on the prime counting function of the general form | π ( x ) − Li ( x ) | < a x ( ln x ) b exp − c ln x for x ≥ x 0 and known constants { a , b , c , x 0 } , we shall show that it is possible to establish exponentially tight effective upper and lower bounds on the prime number theorem. For x ≥ x ∗ , where x ∗ ≤ max { x 0 , 17 } , we have the following: Li ( x ) 1 + a ( ln x ) b + 1 exp − c ln x < π ( x ) < Li ( x ) 1 − a ( ln x ) b + 1 exp − c ln x . These bounds provide a modern, and very clean and explicit, version of the celebrated prime number theorem. Furthermore, it is possible to establish exponentially tight effective upper and lower bounds on the location of the n t h prime. Specifically, we find that p n < Li − 1 n 1 + a ( ln [ n ln n ] ) b + 1 exp − c ln [ n ln n ] for n ≥ n ∗ , whereas p n > Li − 1 n 1 − a ( ln [ n ln n ] ) b + 1 exp − c ln [ n ln n ] for n ≥ n ∗ . Herein, the range of validity is explicitly bounded by some calculable constant n ∗ satisfying n ∗ ≤ max { π ( x 0 ) , π ( 17 ) , π ( ( 1 + e − 1 ) exp 2 ( b + 1 ) c 2 ) } . These bounds provide very clean and up-to-date and explicit information on the location of the n t h prime number. Many other fully explicit bounds along these lines can easily be developed. Overall this article presents a general algorithmic approach to converting bounds on | π ( x ) − Li ( x ) | into somewhat clearer information regarding the primes.