On The Generation Of The Lehmer Problem And Woods Problem In Short Intervals

Let p> 2 be an odd prime. For any integer m, n, the classical Kloosterman sum is defined by where e（y）= e^2Ï€iy, （?） is the multiplicative inverse of a modulo p with a（?）= 1（mod p）.This paper use the estimate of the incomplete Kloosterman sums to study the generation of the D. H. Lehmer problem and A. C. Woods problem. The main content is as the following two aspects:Firstly, let p be an odd prime, H> 0, K> 0, and let I₁^j, I₂^j be subinter-vals of （0,p）,1≤j≤ J, satisfying| for j ≠ k. In this paper, we get thatSecondly, let p be an odd prime, 1≤H≤p,0<δ≤1be any fixed real number, and let I^j be disjoint subintervals of{0,p),1≤ j≤ J, satisfying H/2≤|I^j|≤ H. Define I=∪_i=1^JI^j and A（δ,P） {a∈Z:1≤a,（?）≤p-1,|a-（?）|< δp}, where a is the multiplicative inverse of a modulo p with a（?）= 1（mod p）. Let χ be the Dirichlet character modulo p. We prove and the estimation formula.