What is the jth prime number, pj, for any given positive integer, j?  What are the mathematical processes which generate prime numbers?

Let's conduct an experiment using a tentative algorithm to answer our questions:

Keywords:  Mathematical Analysis, Numerical Analysis,  Riemann prime-counting function ( Rk(x) ), Riemann nontrivial zeta zeros, Prime Number Theorem (PNT), AKS primality test, ..., Mathematica software,  Powerful computer, and Ingenuity!  :-)

Given the index, j =1048, what is jth prime, pj?

We let pj = 3 + Σ ei (from i = 1 to j-1) where ei is a positive even integer.

We assume ei is the distance or gap between two consecutive odd prime numbers.

Note:   2 ≤  ei  ≤  c * log2(pj)  where c > 1 and  as j → ∞, c → 1.

Moreover, we let eavg = log (x) where j = x / log x according to PNT.

(1)  pj_m = x or let  pj_m = j * log (j) (rounded to the nearest prime number where we let m = 0).

If we let pj_m = j * log (j),  then we have  

pj_m = 110,524,084,463,714,192,832,863,589,824,849,481,964,852,871,454,049.

Reference link: 

https://www.wolframalpha.com/input/?i=nearest+prime+10%5E48+log(10%5E48)

Note:   Please see excellent reference link,

https://en.wikipedia.org/wiki/Prime_number_theorem,

for a much better approximation of pj_m.

(2)  Is  j = Rk(pj_m)  as k → ∞ ?

where Rk(x) = R(x) - Σ R(xsl) (from l = k to l = -k), and where

R(x) = Li (x) - Σ 1/n * Li( x1/n )  (from n = 2 to n = ∞)

where Li(x) = ∫ dx'/log(x') (from x' = 2 to x' = x).

Note:  sl  is  the lth  nontrivial zeta zero of the Riemann zeta function.  

If sl  =  .5 + b * i,  then s-l  =  .5 - b * i.

(3)  If the answer to (2) is no, then we have m ← m + 1, and we generate a different pj_m:

If  for (2),  we had j  Rk(pj_m),  then we choose a greater pj_m   appropriately.

Next,  we repeat the processing from (2) to (3) until (2) is true (convergence is achieved).

Reference Links:  

'Why is the stronger Cramér Conjecture by Daniel Shanks and Andrew Granville true?',

https://www.researchgate.net/post/Why_is_the_stronger_Cramer_Conjecture_by_Daniel_Shanks_and_Andrew_Granville_true;

'Uncertainty Principle (UCP) For Predicting Prime Numbers',

https://www.math10.com/forum/viewtopic.php?f=63&t=1484;

' RIEMANN’S PLAN FOR PROVING THE PRIME NUMBER THEOREM', 

http://www.dms.umontreal.ca/~andrew/Courses/Chapter8.pdf;

'Why is the Riemann Hypothesis true?',

https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true

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Romans 8:28, http://biblia.com/verseoftheday/image/Ro8.28

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http://complex.ffn.ub.es/~mbogunya/archivos_cms/files/PhysRevE.90.022806.pdf

http://people.cs.aau.dk/~uk/papers/pgm-book-I-05.pdf

https://image-store.slidesharecdn.com/0e117146-e5b2-4d22-8194-9726cba959d5-original.jpeg

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

https://en.wikipedia.org/wiki/Undecidable_problem

https://www.sciencedaily.com/releases/2015/12/151209142727.htm?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+sciencedaily%2Fcomputers_math%2Fmathematics+%28Mathematics+News+--+ScienceDaily%29

http://arxiv.org/pdf/1502.04573v2.pdf

http://phys.org/news/2016-06-algorithm-random.html

https://en.wikipedia.org/wiki/Prime_number_theorem

http://www.dms.umontreal.ca/~andrew/Courses/Chapter8.pdf

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