Farey sequence

A Farey sequence ( mathematically incorrect also Farey series or just Farey fractions) is in number theory is an ordered set of the shortened breaks out between 0 and 1, the respective denominator as the index does not exceed N. Named are the Farey sequences after the British geologist John Farey.

Formal definition

A Farey sequence N- th order with an ordered set of fractures, with index set and so

Examples

The first 8 episodes in a structured representation:

F1 = {0 · · · · · · · · · · · · · · · · · · · · · 1} F2 = {0 · · · · · · · · · · 1/2 · · · · · · · · · · 1} F3 = {0 · · · · · · 1/3 · · · 1/2 · · · 2/3 · · · · · · 1} F4 = 0 · · · · { 1/4 · 1/3 · · · 1/2 · · · 2/3 · 3/4 · · · · 1} F5 = {0 · · · 1/5 1/4 x 1/ 3 x 2/5 x 1/ 2 x 3/5 x 2 /3 * 3/4 4/5 · · · 1} F6 = 0 · { 1/6 1/5 1/4 x 1/ 3 x 2/5 x 1/ 2 x 3/5 x 2 /3 * 3/4 4/5 5/6 · 1} F7 = {0 x 1/ 7 1/ 6 1/ 5 1/4 2/7 1/3 x 2 /5 3/ 7 1/ 2 4/ 7 3/ 5 x 2/3 5/7 3/4 4 / 5 5/ 6 6/ 7 x 1} F8 = {0 1/8 1/7 1/6 1/5 1/4 2/7 1/3 3/8 2/5 3/7 1/2 4/7 3/5 5/8 2/3 5 / 7 3/ 4 4/ 5 5/ 6 6/ 7 7/ 8 1 } construction

There are at least two ways to construct a Farey series.

Method 1

The first method is first collects all the necessary openings and then sorted. For a Farey sequence and the two fractions and all fractions are used, the denominator q between 2 and N and their counters range from 1 to N-1.

The breaks are for F8 and

All the possible breaks are now reduced as far as possible, sorted in ascending order of size, and deleted duplicate elements:

Method 2

The second method uses a special form of the addition of fractions. The previous Farey sequence must be known to construct the result. It complements the previous Farey sequence cracks that is obtained from an operation each adjacent fractures, but must satisfy the following condition: The sum of the denominators of the two fractions must give N. The operation is as follows: If the two adjacent apertures and, and the sum of the denominator of b and d is N, then the new fracture. For this operation, the name Farey addition has established. By -made restriction applies to any Farey sequence that it is a subset of the Peirce - numbers.

Is assumed, a recursive construction is possible.

Example

To be calculated. itself is yet to be created assumed to be known, or. With adjacent fractions, the denominator sum is equal to 7, the new elements are formed by addition of the numerator and denominator:

The new elements are:

Properly sorted arises now

Properties

The cardinality of a Farey sequence for index N is equal to the cardinality of the previous result to the index N-1 added to the value of Euler's φ - function for N:

For two consecutive fractures and a Farey sequence the products a · b · c and d give two consecutive numbers. One can also write:

Are reversed and two fractions with and so are neighbors to the Farey sequence, in other words, each intermediate fraction has a denominator. In fact, namely the need to counter the positive fractions, and positive integers be so and.

It follows

Likewise follows

Both inequalities are sharp for exactly the Farey sum.

Farey sequences and Riemann conjecture

Jerome Franel proved 1924 ( supplemented by Edmund Landau ) that the Riemann conjecture is a statement about Farey series equivalent.

Are the elements of the n-th Farey sequence and is the distance between the k-th term of the n-th and the k-th Fareyfolge term of the series of points equidistant in the unit interval with the same number of terms, such as the n-th Fareyfolge. ( Use the Landau symbols) Franel then proved the equivalence of the Riemann hypothesis:

And Landau noted that the Riemann hypothesis then also to

Equivalent.