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- Landau_prime_ideal_theorem abstract "In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.What to expect can be seen already for the Gaussian integers. There for any prime number p of the form 4n + 1, p factors as a product of two Gaussian primes of norm p. Primes of the form 4n + 3 remain prime, giving a Gaussian prime of norm p2. Therefore we should estimatewhere r counts primes in the arithmetic progression 4n + 1, and r′ in the arithmetic progression 4n + 3. By the quantitative form of Dirichlet's theorem on primes, each of r(Y) and r′(Y) is asymptoticallyTherefore the 2r(X) term predominates, and is asymptoticallyThis 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 proved in Landau 1903, for norm at most X the same asymptotic formulaalways holds. Heuristically this is because the logarithmic derivative of the Dedekind zeta-function 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. The number of prime ideals of norm ≤ X iswhere cK is a constant depending on K.".
- Landau_prime_ideal_theorem wikiPageID "3228360".
- Landau_prime_ideal_theorem wikiPageRevisionID "474653716".
- Landau_prime_ideal_theorem hasPhotoCollection Landau_prime_ideal_theorem.
- Landau_prime_ideal_theorem subject Category:Theorems_in_algebraic_number_theory.
- Landau_prime_ideal_theorem subject Category:Theorems_in_analytic_number_theory.
- Landau_prime_ideal_theorem type Abstraction100002137.
- Landau_prime_ideal_theorem type Communication100033020.
- Landau_prime_ideal_theorem type Message106598915.
- Landau_prime_ideal_theorem type Proposition106750804.
- Landau_prime_ideal_theorem type Statement106722453.
- Landau_prime_ideal_theorem type Theorem106752293.
- Landau_prime_ideal_theorem type TheoremsInAlgebraicNumberTheory.
- Landau_prime_ideal_theorem type TheoremsInAnalyticNumberTheory.
- Landau_prime_ideal_theorem comment "In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.What to expect can be seen already for the Gaussian integers. There for any prime number p of the form 4n + 1, p factors as a product of two Gaussian primes of norm p. Primes of the form 4n + 3 remain prime, giving a Gaussian prime of norm p2.".
- Landau_prime_ideal_theorem label "Landau prime ideal theorem".
- Landau_prime_ideal_theorem sameAs m.08_tr5.
- Landau_prime_ideal_theorem sameAs Q6484331.
- Landau_prime_ideal_theorem sameAs Q6484331.
- Landau_prime_ideal_theorem sameAs Landau_prime_ideal_theorem.
- Landau_prime_ideal_theorem wasDerivedFrom Landau_prime_ideal_theorem?oldid=474653716.
- Landau_prime_ideal_theorem isPrimaryTopicOf Landau_prime_ideal_theorem.