The nth Prime Exponentially

Consider both the Logarithmic integral, Li(x)=limϵ→0∫01−ϵdulnu+∫1+ϵxdulnu, and the prime counting function π(x)=∑p≤x1. From several recently developed known effective bounds on the prime counting function of the general form |π(x)−Li(x)|<ax(lnx)bexp−clnx for x≥x0 and known constants {a,b,c,x0}, 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{x0,17}, we have the following: Li(x)1+a(lnx)b+1exp−clnx<π(x)< Li(x)1−a(lnx)b+1exp−clnx. 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 nth prime. Specifically, we find that pn<Li−1n1+a(ln[nlnn])b+1exp−cln[nlnn] for n≥n∗, whereas pn>Li−1n1−a(ln[nlnn])b+1exp−cln[nlnn] for n≥n∗. Herein, the range of validity is explicitly bounded by some calculable constant n∗ satisfying n∗≤max{π(x0),π(17),π((1+e−1) exp2(b+1)c2)}. These bounds provide very clean and up-to-date and explicit information on the location of the nth 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.