Dirichlet's Theorem

The Dirichlet theorem on arithmetic progressions or Dirichlet prime number theorem states that there are infinitely many primes of the form a + nd, where n = 1,2,3,4,..... is also a positive integer, for any two positive coprime integers a and d. In other words, there is an infinite number of primes that are modulo d congruent. The theorem extends Euclid's theorem that there are infinitely many prime numbers, and is named after Peter Gustav Lejeune Dirichlet.

The arithmetic progression of numbers of the form a + nd is known as Dirichlet's theorem, which states that this series includes an infinite number of prime numbers. Stronger versions of Dirichlet's theorem state that the sum of the reciprocals of the prime numbers in the progression diverges for every such arithmetic progression, and that different such arithmetic progressions of the same modulus have roughly the same proportions of primes. The primes are distributed evenly among the congruence classes modulo d that contain a's coprime to d.

Peter Gustav Lejeune Dirichlet proved the Dirichlet theorem on arithmetic progressions by using Dirichlet L-series. The value of the Dirichlet L-function of a non-trivial character at 1 is nonzero, proving Dirichlet's theorem. This argument can be proved using calculus and analytic number theory. The primes that are congruent to 1 modulo some ‘n’ in the particular case a = 1 can be proved without using calculus by studying the splitting action of primes in cyclotomic extensions. The prime number theorem states that primes thin out on average, so the same must be true for primes in arithmetic progressions. For a given value of d, it's natural to inquire about how primes are shared among the various arithmetic progressions. Euler's totient function (d) gives the number of feasible progressions modulo d where a and d do not have a common factor > 1. As compared to progressions with a quadratic residue remainder, progressions with a quadratic nonresidue remainder usually have slightly more components.

Dirichlet's Unit Theorem

Dirichlet's unit theorem, named after Peter Gustav Lejeune Dirichlet, is a fundamental result in algebraic number theory. It defines the rank of a group of units in a number field K's ring OK of algebraic integers. The regulator is a positive real number that controls the "density" of the units.

The statement is that the group of units is finitely generated and has a rank (maximal number of multiplicatively independent elements) of r = r1 + r2 - 1 where r1 is the number of real embeddings and r2 is the number of conjugate pairs of complex embeddings of K.

This definition of r1 and r2 is based on the assumption that there will be as many ways to embed K in the complex number field as the degree n = [K : Q]; these will either be into real numbers or pairs of embeddings linked by complex conjugation, resulting in n = r1 + 2r2.

It's worth noting that if K is Galois over Q, either r1=0 or r2=0.

The Two Other Ways to Find the Value of r1 and r2 are as Follows:

• Using the primitive element theorem to write K = Q(ɑ), and then r1 is the number of real conjugates of and 2r2 is the number of complex conjugates. In other words, if f is the minimal polynomial of over Q, then r1 is the number of real roots, and 2r2 is the number of non-real complex roots of f that occur in complex conjugate pairs.

• As a product of fields, write the tensor product of fields K ⊗Q ℝ, with r1 copies of R and r2 copies of C.

Ex: The rank of a quadratic field is 1 if it is a true quadratic field and 0 if it is an imaginary quadratic field.

Here the theory of Pell's equation is the theory for real quadratic fields.

Except for Q and imaginary quadratic fields, which have rank 0, all number fields have a positive rank. A determinant known as the regulator is used to determine the size of the units in general. In theory, a basis for the units can be efficiently computed; however, when n is large, the calculations become very complex.

The set of all roots of unity of K that form a finite cyclic group is the torsion in the group of units. As a result, the torsion of a number field with at least one real embedding must be just 1,1. There are number fields with no real embeddings that also have 1,1 for the torsion of their unit group, such as most imaginary quadratic fields.

In terms of units, completely real fields are special. If the unit groups for the integers of L and K have the same rank and L/K is a finite extension of number fields with a degree greater than 1, then K is totally real and L is a totally complex quadratic extension. The opposite is also true. The theorem holds for every order O ⊂ OK, not just the maximal order OK. 

Helmut Hasse and Claude Chevalley developed a generalization of the unit theorem to define the structure of the group of S-units, deciding the rank of the unit group in localization of rings of integers. Q ⊕ OK, S ⊗Z Q Galois module structure has also been determined.

Dirichlet's Approximation Theorem

Dirichlet's theorem on Diophantine approximation, also known as Dirichlet's approximation theorem, states that for any real numbers ɑ and N, with 1Nthere, exist integers p and q such that 1≤ q ≤ N and|qα- p|≤ \[\frac{1}{[N]+1}\] < \[\frac{1}{N}\]

The integer part of N is represented by [N]. This is a fundamental result in Diophantine approximation, demonstrating that every real number has a sequence of good rational approximations: in fact, the inequality is fulfilled by infinitely many integers p and q for any given irrational α |α - \[\frac{p}{q}\]| < \[\frac{1}{q^{2}}\]

The Thue–Siegel–Roth theorem, a result in the opposite direction, provides basically the tightest possible limit, in the sense that the bound on the rational approximation of algebraic numbers cannot be strengthened by increasing the exponent beyond 2.

Conclusion

When a and d are relatively prime and n runs over the positive integers, Dirichlet's theorem states that there is an infinite number of primes in an arithmetic progression a + nd. Although certain special cases of Dirichlet's theorem, such as the arithmetic progression 2 + 3n, can be proved using simple methods, the general case is much more difficult to prove. The Riemann zeta-function and Dirichlet L-functions are used in the analysis. The Dirichlet theorem is often used to prove that a prime number exists that meets a certain congruence condition while preventing a finite number of bad primes. It allows us to find the density of the set of primes at which a finite set of integers has prescribed Legendre symbol values in a more general sense.