Exponential Sums Involving The Divisor Function Over Arithmetic Progressions

The divisor functions are the basic arithmetic functions in number theory.Hence the behavior of the divisor functions are very important in the theory of automorphic L-funcions.In this thesis,we consider the oscillation behavior of the divisor function and the exponential function in arithmetic progressions.Let φ(x)be a smooth function of compact support in(0,∞)with derivatives bounded by φ(j)(x)<<1 and d3(n)be the number of ways to write n as a product of three factors.We get the asymptotic formula for the nonlinear exponential sum In this thesis,we can get two theorems.In Theorem 1,let k,l,q∈N.Then for any ?>0 and q<<X1/3-?,we have where c(φ)=∫0∞uφ(u3)du.Under the conditions of Theorem 1,let q be a prime number,and we can obtain the more precise result.