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The high-dimensional D. H. Lehmer problem over quarter intervals is studied. By using the properties of character sum and the estimates of Dirichlet L-function, the previous result is improved to be the best possible in the case of q = p, an odd prime with p≡1(mod 4), which is shown by studying the mean square value of the error term

High-Dimensional D. H. Lehmer Problem over Quarter Intervals