THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS

Abstract

Let α(n) denote the Fourier coefficients of cusp forms or the number of divisors of n. Estimates of the type b(q) n≤X n≡b(q) α(n) − main term ∣∣∣2 ε X1+ε 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≤X α(n)ψ(n) for any q-periodic function ψ can be estimated non-trivially.