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Let $F(x) = \sum_{1 < n \leq x} (-1)^{\pi (n)}$ where $\pi (n) $ is the prime counting function.

I am trying to understand how $ F(x) $ behaves as $ x \to \infty$. In particular, what are the asymptotics for $F$? I can't seem to use any tools that handle sums of the form, $\sum_{n \leq x} f(n)$ where $f:\mathbb{N} \to \mathbb{C}$. Does anyone have any suggestions?

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This might be an open problem. Have you checked Guy's "Unsolved Problems in Number Theory"? – Greg Martin Dec 6 '12 at 10:22
I suspect knowing these asymptotics would shed some light on the prime k-tuples conjecture as well as several other conjectures concerning prime gaps. It is easy with known results to get something like size of F(n) is often bigger than logn G(n), where G(n) is a slow growing ratio of iterated logs, but solving this posted problem is morally (if not tautologically) equivalent to establishing exact bounds on the prime gap function. Gerhard "It Looks Tautological To Me" Paseman, 2012.12.06 – Gerhard Paseman Dec 6 '12 at 17:20
For some mystical amusement, I computed the first few inflection values of F and submitted some of the consecutive positive values (3,1,5,3,7,1,3) to OEIS and Google search. Quite a few sequences have this pattern, a near match being ruler markings. Gerhard "Is It Coincidence, Or Murder?" Paseman, 2012.12.06 – Gerhard Paseman Dec 6 '12 at 17:40
Now that I have spent some time thinking about it, your series is equivalent (but not equal) to the alternating sum of primes, for which some literature exists. I leave the relevant searches to you. Good Luck. Gerhard "Ask Me About System Design" Paseman, 2012.12.06 – Gerhard Paseman Dec 6 '12 at 19:58
Greg, I am trying to solve a problem in Guy's book. Gerhard, do you know if it is true that $F(n) = O(n^{\alpha})$ where $0 < \alpha < 1 $? – Mustafa Said Dec 7 '12 at 4:34

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