The divisor function over arithmetic progressions

provided x is sufficiently large. An asymptotic formula of type (1) Df (x; q, a) = (1 +O((log x)))Df (x; q) , in which the error term is smaller than the main term by a suitable power of log x, is good enough for basic applications. More important than the size of the error term is the range where (1) holds uniformly with respect to the modulus q. In this paper we consider the problem for the divisor function f(n) = τ(n). In this case one can prove by a simple elementary argument that ∆f (x; q, a) = Df (x; q, a) −Df (x; q) ≪ x , which yields (1) in the range q < x. Using Fourier series technique and Weil’s estimate for Kloosterman sums