Abstract: For a positive integer m, a finite set of integers is said to be evenly distributed modulo m if the set contains an equal number of elements in each congruence class modulo m. This paper mainly studies the condition of m when the multiplicity of numerical semigroup is 4, and through the definition of numerical semigroup and related lemmas, it is proved that m satisfies when the gap set of numerical semigroups is evenly distributed.

Key words: numerical semigroup, gap, multiplicity, embedding dimension