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We show that an inequality, proven by Küronya-Pintye, which governs the behavior of the log-canonical threshold of an ideal over P n and that of its Castelnuovo-Mumford regularity, can be applied to the setting of principally polarized abelian varieties by substituting the Castelnuovo-Mumford regularity with Θ-regularity of Pareschi-Popa. this we can give a new proof of a celebrated result of Ein-Lazarsfeld on the singularities of pluritheta-divisors. Then we prove an upper bound for the Θ-regularity index of multiplier ideals, by following a similar argument to the one of [6]. This together with Nadel Vanishing will allow us to conclude. We end the paper by computing some examples. The firs one, that was suggested to us by Z. Jiang, shows that the inequality provided is sharp.Notation and conventions. We work over the field of the complex numbers. Given a morphism µ : X → X of smooth projective varieties, we denote the relative canonical divisor bywhere K X and K X stand for the canonical classes on X and X respectively. The round up and round down of rational numbers and Q-divisors are denoted by ⌈·⌉ and ⌊·⌋.