Sum of Divisors in a Ring of Gaussian Integers

Abstract

We construct an asymptotic formula for a sum function for σ _a (α), where σ _a (α) is the sum of the ath powers of the norms of divisors of the Gaussian integer α on an arithmetic progression α ≡ α₀ (mod γ) and in a narrow sector ϕ₁ ≤ arg α < ϕ₂. For this purpose, we use a representation of σ _a (n) in the form of a series in the Ramanujan sums.