HOME

TheInfoList



OR:

In
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, the prime ideal theorem is the
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
generalization of the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
. It provides an asymptotic formula for counting the number of
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s of a number field ''K'', with
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
at most ''X''.


Example

What to expect can be seen already for the
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s. There for any prime number ''p'' of the form 4''n'' + 1, ''p'' factors as a product of two
Gaussian prime In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathb ...
s of norm ''p''. Primes of the form 4''n'' + 3 remain prime, giving a Gaussian prime of norm ''p''2. Therefore, we should estimate :2r(X)+r^\prime(\sqrt) where ''r'' counts primes in the arithmetic progression 4''n'' + 1, and ''r''′ in the arithmetic progression 4''n'' + 3. By the quantitative form of
Dirichlet's theorem on primes 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'' is al ...
, each of ''r''(''Y'') and ''r''′(''Y'') is asymptotically :\frac. Therefore, the 2''r''(''X'') term predominates, and is asymptotically :\frac.


General number fields

This general pattern holds for number fields in general, so that the prime ideal theorem is dominated by the ideals of norm a prime number. As
Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopol ...
proved in , for norm at most ''X'' the same asymptotic formula :\frac always holds. Heuristically this is because the
logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula \frac where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f' ...
of the
Dedekind zeta-function In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
of ''K'' always has a simple pole with residue −1 at ''s'' = 1. As with the Prime Number Theorem, a more precise estimate may be given in terms of the
logarithmic integral function In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
. The number of prime ideals of norm ≤ ''X'' is : \mathrm(X) + O_K(X \exp(-c_K \sqrt)), \, where ''c''''K'' is a constant depending on ''K''.


See also

* Abstract analytic number theory


References

* * * {{cite book , author=Hugh L. Montgomery , authorlink=Hugh Montgomery (mathematician) , author2=Robert C. Vaughan , authorlink2=Robert Charles Vaughan (mathematician) , title=Multiplicative number theory I. Classical theory , series=Cambridge tracts in advanced mathematics , volume=97 , year=2007 , isbn=978-0-521-84903-6 , pages=266–268 Theorems in analytic number theory Theorems in algebraic number theory