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Landau prime ideal theorem : ウィキペディア英語版
Landau prime ideal theorem
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''.
==Example==
What to expect can be seen already for the Gaussian integers. There for any prime number ''p'' of the form 4''n'' + 1, ''p'' factors as a product of two Gaussian primes 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, each of ''r''(''Y'') and ''r''′(''Y'') is asymptotically
:\frac.
Therefore the 2''r''(''X'') term predominates, and is asymptotically
:\frac.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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