On The Generalization Of The D.H.Lemher Problem In Unions Of Short Intervals

Let p>2 be an odd prime.For any integer m,n,the classical Kloosterman sum is defined by (?) where e(y)= e2Tπiy,a is the multiplicative inverseof a modulo,,satisfying 1 ≤a ≤ p-1 and aa =1(mod p).Let p be an odd prime,x denote any Dirichlet character mod p.Assume that m,n,k are integers with k≥ 2,the two-term exponential sum C(m,n,k;P)is defined as follows:(?)Firstly,let p be an odd prime,H>0,K>0,and let I1(j),I2(j)be subin-tervals of(0,p),1≤j≤J,satisfying |I1(j)|=H,|I2(j)|=K,and I1(j)∩I1(k)=φfor j ≠ k.Assume that c,n are integers with n≥2 and(n,P)=(c,p)1.We prove that(?)Secondly,let p be an odd prime,H>0,K>0,and let I1(j),I2(j)be subin-tervals of(0,p),1 ≤j≤ J,satisfying |I1(j)|= H,|I2(j)|= K,and I1(j)∩I1(k)=φfor j ≠k.Assume that c,n,l are positive integers,n≥2 with(nc,p)=1 and l|n.We prove that (?) where φ(q)is the Euler function,and ω(q)denotes the number of distinct prime factors of q.Thirdly,let p be an odd prime,1≤H≤p,0<δ<1 be any fixed real number,and m≥ 2 be integers.Let I(j)be disjoint subintervals of(0,p),1≤ j ≤J,satisfying H/2 ≤|I(j)|≤ H,and let(y)p denote the non-negative least residue of y modulo p.Define I = ∪j=1 J I(j),and let χ be the Dirichlet character modulo p.We prove that (?) and (?)...