THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS

Let α(n) denote the Fourier coefficients of cusp forms or the number of divisors of n. Estimates of the type$\sum^*_{b(q)}|\sum_{n\leq X n\equiv b(q)}\alpha(n)-\rm main term |^2\ll_{\varepsilon} X^{1+\varepsilon}$are shown, uniformly in q ≤ X. The methods can be extended to other arithmetic functions, for example, the number of representations of n as a sum of two squares or k-free numbers. As an application, sums of the type ∑n ≤ Xα(n) ψ(n) for any q-periodic function ψ can be estimated non-trivially.