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.

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


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.


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