On the Multiplicity of a Proportionally Modular Numerical Semigroup - Ze Gu (Discrete Dynamics in Nature and Society)
A proportionally modular numerical semigroup is the set Sa,b,c of nonnegative integer solutions to a Diophantine inequality of the form ax mod b≤cx, where a,b, and c are positive integers. A formula for the multiplicity of Sa,b,c, that is, mSa,b,c=kb/a for some positive integer k, is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for k. Furthermore, we obtain k=1 for various cases and a formula for the number of the triples a,b,c such that k≠1 when the number a−c is fixed.