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

Abstract

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 $p\equiv 1$(mod 4), which is shown by studying the mean square value of the error term.