There is not much here except a link to my incomplete and incorrect proof.

But, the proof such as it is does give an indication of how I have used the Fouries Series to transform the sifting functions of the Sieve of Eratothenes from discreet integer function into discontinuous, but real functions. This alows one to bring to bear the power of a real and complex analysis to the problem.

But, as is the bane of all such sieve approaches to prime number distribution, I have yet to prove the error term has an acceptably small bound. I am sure the error term is bounded by the number of strands in the sieve. But am unable at this time to prove this conjecture.

Here is a a paper I recently posted tot he arXiv pre-print archive on the Auto and Cross Correlation of Ramanujan Sums