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:

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

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.

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.

Wikipedia's article on Farey Fractions points to an article of Jerome Framel that some averages are requivalent 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:

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

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.

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

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:

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

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.