Type I information at level X^{1/2} for product-form weights in Q(√−n)

Establish Type I estimates at level L = X^{1/2} for the Type I sum defined by ∑_{Nd ∼ L} |∑_{d | a, N a ≤ X} w(a)| in the imaginary quadratic field K = Q(√−n), where the weight function w is of product form w = f ⊠_ℓ f′ as in Definition 2.1 and f, f′ are supported on [±2n X^{1/2}]. In particular, determine whether analogous bounds at level L = X^{1/2} can be obtained without the absolute values in the inner sum, namely for ∑_{Nd ∼ L} ∑_{d | a, N a ≤ X} w(a), corresponding to Type I₂ information.

Background

In their sieve framework over the number field K = Q(√−n), the authors rely on Type I and Type II information to control sums over prime ideals associated with primes of the form x^2 + n y^2. Type I sums are defined by equation (3.1) as ∑{Nd ∼ L} |∑{d | a} w(a)|, with L referred to as the level. Achieving sufficiently high levels for Type I and Type II estimates is crucial to run the Duke–Friedlander–Iwaniec sieve and derive asymptotics.

The paper proves Type I information up to level L = X^{1/2 − o(1)} and Type II up to L between X^{o(1)} and X^{1/2 − o(1)}, which suffices for their main theorem. However, they highlight that reaching the critical level L = X^{1/2} for Type I would be a substantive advance. Moreover, even proving bounds at that level without absolute values in the Type I sum (a form of Type I₂ information) appears difficult; achieving this would enable arbitrary logarithmic savings in their main asymptotic for primes of the form x^2 + n y^2.

This open problem is specifically about product-form weights w = f ⊠_ℓ f′ (Definition 2.1) tied to their Gaussian and number field setting, and directly concerns improving the sieve inputs beyond what is currently established in the paper.