Dirichlet density
Encyclopedia
In mathematics
, the Dirichlet density (or analytic density) of a set of primes
, named after Johann Gustav Dirichlet
, is a measure of the size of the set that is easier to use than the natural density
.
is the limit
if the limit exists. This expression is usually the order of the "pole" of
at s = 1, (though in general it is not really a pole as it has non-integral order), at least if the function on the right is a holomorphic function times a (real) power of s−1 near s = 1. For example, if A is the set of all primes, the function on the right is the Riemann zeta function which has a pole of order 1 at 0, so the set of all primes has Dirichlet density 1.
More generally, one can define the Dirichlet density of a sequence of primes (or prime powers), possibly with repetitions, in the same way.
/(number of primes less than N)
then it also has a Dirichlet density, and the two densities are the same.
However it is usually easier to show that a set of primes has a Dirichlet density, and this is good enough for many purposes. For example, in proving Dirichlet's theorem on arithmetic progressions
, it is easy to show that the Dirichlet density of primes
in an arithmetic progression a + nb (for a, b coprime) has Dirichlet density 1/φ(b), which is enough to show that there are an infinite number of such primes, but harder to show that this is the natural density.
Roughly speaking, proving that some set of primes has a non-zero Dirichlet density usually involves showing that certain L-functions do not vanish at the point s = 1, while showing that they have a natural density involves showing that the L-functions have no zeros on the line Re(s) = 1.
In practice, if some "naturally occurring" set of primes has a Dirichlet density, then it also has a natural density, but it is possible to find artificial counterexamples: for example, the set of primes whose first decimal digit is 1 has no natural density, but has Dirichlet density log(2)/log(10).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Dirichlet density (or analytic density) of a set of primes
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, named after Johann Gustav Dirichlet
Johann Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...
, is a measure of the size of the set that is easier to use than the natural density
Natural density
In number theory, asymptotic density is one of the possibilities to measure how large a subset of the set of natural numbers is....
.
Definition
If A is a subset of the prime numbers, the Dirichlet density of Ais the limit
if the limit exists. This expression is usually the order of the "pole" of
at s = 1, (though in general it is not really a pole as it has non-integral order), at least if the function on the right is a holomorphic function times a (real) power of s−1 near s = 1. For example, if A is the set of all primes, the function on the right is the Riemann zeta function which has a pole of order 1 at 0, so the set of all primes has Dirichlet density 1.
More generally, one can define the Dirichlet density of a sequence of primes (or prime powers), possibly with repetitions, in the same way.
Properties
If a subset of primes A has a natural density, given by the limit of/(number of primes less than N)
then it also has a Dirichlet density, and the two densities are the same.
However it is usually easier to show that a set of primes has a Dirichlet density, and this is good enough for many purposes. For example, in proving Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other words, there are infinitely many primes which are...
, it is easy to show that the Dirichlet density of primes
in an arithmetic progression a + nb (for a, b coprime) has Dirichlet density 1/φ(b), which is enough to show that there are an infinite number of such primes, but harder to show that this is the natural density.
Roughly speaking, proving that some set of primes has a non-zero Dirichlet density usually involves showing that certain L-functions do not vanish at the point s = 1, while showing that they have a natural density involves showing that the L-functions have no zeros on the line Re(s) = 1.
In practice, if some "naturally occurring" set of primes has a Dirichlet density, then it also has a natural density, but it is possible to find artificial counterexamples: for example, the set of primes whose first decimal digit is 1 has no natural density, but has Dirichlet density log(2)/log(10).