First I'll toot my own horn.
There is still some work left for elementary and near elementary methods to accomplish. Based on your description, I think your formulas say something about the distribution of numbers coprime to the kth primorial. I have been working on something similar, and part of the path has led me to finding some elementary arguments which improve on part of the literature. I tell some of the story in the MathOverflow question http://mathoverflow.net/questions/37679/erik-westzynthiuss-cool-upper-bound-argument-update . The question title refers to a nice argument which serves as an introduction to sieve theory, and shows the potential for getting a handle on something as unwieldy as the distribution of primes which has a lot of regular and fractal behaviour occuring in its development.
If you are interested in this type of mathematics, you could do worse than to read through that question and the answers to it. If you want to know about general lower bound results, the Westzynthius paper has a nice construction which will produce gaps between primes which are larger than average, also using elementary means; it was the first published construction to show that for any constant C there are infinitely many k such that $p_{k+1} \gt C \log p_k + p_k$. You might even find a way to make elementary improvements on the arguments, as well as search the literature to find improvements by Rankin, Erdos, and others. (If you are patient, you can wait for a writeup I am doing which includes an interpretation of the key results of the paper.)
I am somewhat interested in the result of yours, but I suspect that I speak for others as well as myself when I say I would prefer a single approximation (or a small system of equations that I could numerically compute) to an exponential family of formulas I would need to calculate one value precisely. I believe that is one advance of sieve theory over elementary methods: it caters to such a preference. I don't know of any very accessible literature on the subject, but I sometimes refer to Cojocaru and Murty's book, and those more familiar with the literature may come with their recommendations. If you and I are fortunate, we may hear from the likes of zeb or quid, whose opinions I believe are more informed than mine on this subject.
Disclaimer: my training is not in analytic number theory; I suspect my perspective on the subject is much the same as yours.
Gerhard "But That Doesn't Stop Me" Paseman, 2012.01.17
