Farey Fractions Estimate Equivalent to the Prime Number Theorem?

Question

Wikipedia's article on Farey Fractions points to an article of Jerome Franel that some averages are equivalent to the Riemann hypothesis.

Let $F_n$ be the $n$-th Farey sequence, then the number of elements in that list is:

$$|\{ (m,n): 0 \leq m, n \leq N \text{ and }\mathrm{gcd}(m,n)=1\}| = \phi(1) + \dots + \phi(N) $$

Then traversing the Farey sequence in order, we can estimate:

$$ \sum_{k = 1}^{|F_n|} \left(a_k - \frac{k}{|F_n|} \right)^2 = O(n^r) \quad\text{for all } r>-1$$

which is claimed to be equivalent to the Riemann hypothesis (any proof?).

I interpret this as measuring the discrepancy (see also this book) between the Farey sequence and a uniformly spaced sequence.

There are lost of statements equivalent to the Riemann Hypothesis, but it is far from proven. Is there a similar or weaker estimate which could be equivalent of the to the Prime Number Theorem? One equivalent statement is that

$$ \frac{1}{x} \sum_{n \leq x} \Lambda(n) = 1 + o(1)$$

In fact, it seems that estimates on the set of primes, control and benchmark our ability to estimate other number-theoretic quantities. So I am wondering if there is an estimate of Farey fractions that captures similar information.

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Franel's paper doesn't seem to be on the internet and I don't have access to a University library.

Answer 1

I am surprised no one answered this, I just ran across it randomly when I was looking for something else. I have read Franel's paper, and Landau's, which is the next one in the same journal (if you're getting one, you might as well get both). However, those papers are in a box somewhere and I can't find them at the moment. The answer is that yes, the Prime Number Theorem is known to be equivalent to the little-oh estimate $$\sum_{k=1}^{|F_n|}\left(a_k-\frac{k}{|F_n|}\right)^2=o(1).$$ I think this is proved in Franel's paper. To see that the bound implies the PNT is fairly easy, but to show the reverse implication is not that easy.

A more accessible reference to add to the list already mentioned is Chapter 9 of "The distribution of prime numbers," by M.N. Huxley.