On numbers of the form n = (u^2 + dv^2)ω in arithmetic progression

Abstract

Let us R(n) denotes the number of representations of positive integers n by form n = (u2 + v2)ω, u,v ∈ Z, ω ∈ N. The function R(n) is an analogue of the divisor function d3(n). Summarize the Heath-Brown results on distribution of value of the divisor function d3(n) on an arithmetical progression n ≡ a(modq), (a, q) = 1, with increasing the arithmetical ratio together with x, an asymptotic formula for summatory function for R(n) was being construct, which is a non-trivial for q → ∞. The proof of this result use the truncated functional equation on the line Res = 1/2 + Δ, │Δ│ < 1/2 of the Hecke Zeta function with transport of an imaginary quadratic field Q(√-d).