UNIFORM BOUNDS OF THE PILTZ DIVISOR PROBLEM OVER NUMBER FIELDS

We consider the upper bound of the Piltz divisor problem over number fields. The Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this function for many cases. Our proof uses the estimate of exponential sums. We also show uniform results for the ideal counting function and relatively r-prime lattice points as one of its applications.