Average reciprocals of the order of a modulo n

Let a>1a>1 be an integer. Denote by la(n)la(n) the multiplicative order of a modulo integers n. We prove that∑n≤x,(n,a)=11la(n)=Oa(xexp⁡(−(12+o(1))log⁡xlog⁡log⁡log⁡xlog⁡log⁡x)), which is an improvement over [18, Theorem 5].Further, we obtain several applications toward number fields and 2-dimensional abelian varieties of CM-type.