The coefficients of automorphic <i>L</i>-function on GLm in short intervals

Let [Formula: see text] denote the coefficients of an automorphic L-function on [Formula: see text]. Let [Formula: see text] be a multiplicative function which satisfies the following conditions: For every prime number p outside of an exceptional set S, [Formula: see text], where B is a non-zero integer. Also for every positive integer n, [Formula: see text], where [Formula: see text] is the [Formula: see text]-dimensional divisor function. Moreover, the exceptional set S satisfies [Formula: see text] for any [Formula: see text]. Then under GRH and GRC, we get a squareroot cancelation for [Formula: see text] in almost all short intervals, provided [Formula: see text] as [Formula: see text], [Formula: see text] and [Formula: see text] for any fixed [Formula: see text]. Let f be an even weight [Formula: see text], level 1 primitive, Hecke-normalized holomorphic cusp form without complex multiplication. For the symmetric power L-functions [Formula: see text] we only need to assume GRH. And we find that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are within the scope of our results, which generalize the work of Chinis. In the course of the proof we unconditionally obtain an upper bound on the summation of [Formula: see text] over smooth numbers.