Abstract. We prove an asymptotic formula for a variant of the binary additive divisor problem with linear factors in the arguments, which has a power saving error term and which is uniform in all involved parameters.

We prove an asymptotic formula for the shifted convolution of the divisor functions d 3 (n) and d(n), which is uniform in the shift parameter and which has a power-saving error term. The method is also applied to give analogous estimates for the shifted convolution of d 3 (n) and Fourier coefficents of holomorphic cusp forms. These asymptotics improve previous results obtained by several different authors.for any ε > 0, where P is a polynomial of degree three.Bykovskiȋ and Vinogradov [2] returned to the spectral approach of Deshouillers [3] based on the Kuznetsov formula and stated (1.1) with an exponent 8 9 in the error term. Unfortunately, not more than a few brief hints were given to support this claim, and our first result is a detailed proof of the following asymptotic formula, which yields in addition a substantial range of uniformity in the shift parameter h. Theorem 1.1. We have for h ≪ x 2 3 , D ± (x; h) = xP h (log x) + O x 8 9 +ε ,

We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line.

Given a multiplicative function f which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum |h|