The Landau prime ideal theorem, proved by Edmund Landau in 1903, is the number field generalization of the prime number theorem. It gives an asymptotic formula for the number of prime ideals of a number field with norm at most X, showing that count is asymptotically X over the natural logarithm of X, matching the classical prime-counting pattern even though primes factor differently across different number fields. The result follows from the fact that the Dedekind zeta function of a number field always has a simple pole with residue negative one at s equals one. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementAn asymptotic formula for the number of prime ideals of a number field K with norm at most X, namely X/log X. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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Source Landau prime ideal theorem (Wikipedia)
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1. Landau prime ideal theorem (Wikipedia)
Introduction
It provides an asymptotic formula for counting the number of prime ideals of a number field K, with norm at most X.
General number fields
As Edmund Landau proved in Landau 1903
In Branch: Algebraic Number Theory, Lead sentence
In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem.
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