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Let a>1 be an integer. Denote by la(p) the multiplicative order of a modulo primes p. We prove that if xlogxloglogx=o(y), then1y∑a≤y∑p≤x1la(p)=logx+Cloglogx+O(1)+O(xyloglogx), which is an improvement over a theorem by Felix [Fe]. Additionally, we also prove two other related average results.
Let a>1 be an integer. Denote by la(n) the multiplicative order of a modulo integers n. We prove that∑n≤x,(n,a)=11la(n)=Oa(xexp(−(12+o(1))logxlogloglogxloglogx)), which is an improvement over [18, Theorem 5].Further, we obtain several applications toward number fields and 2-dimensional abelian varieties of CM-type.
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