Long Dirichlet-Polynomial Recipe for Quadratic L-Functions

Definitions and mathematical statement

For a fundamental discriminant $d$, let $\chi_d$ denote its associated quadratic Dirichlet character. The supplied record concerns the finite Dirichlet polynomial \[ P_A(X;d)=\sum_{1\leq n\leq X}\frac{\tau_A(n)\chi_d(n)}{\sqrt n}. \] Here $A$ and the coefficients $\tau_A(n)$ are notation retained from the source; the record does not define them or specify their admissible values.

The proposed long-polynomial recipe asserts, schematically, that \[ \text{a smoothed average of }P_A(X;d)\text{ over fundamental discriminants }d = \text{an integral of quadratic }L\text{-function swap terms} +O\!\left(D^{1-\delta}\right) \qquad(D\to\infty). \] This display records the advertised target, not a fully specified mathematical conjecture. The supplied record omits the averaging weight and normalization, the discriminant range and sign restrictions, the relation between $X$ and $D$, the definition and admissible shifts of $A$, the swap terms, and the integral, including its contour or domain. It also omits the quantification and parameter dependence of $\delta$, the uniformity of the error estimate, and the permitted dependence of its implied constant. A positive $\delta$ would express a power saving, but the required hypotheses for such a saving are not supplied. These omissions prevent an exact quantified formulation.

The supplied dated literature assessment describes a conditional result of Conrey--Rodgers in a polynomial-length range with exponent below $2$, corresponding to the $0$- and $1$-swap terms. It describes the longer range and the full shifted-moment prediction as remaining open. The conditioning hypotheses, definition of that exponent, and full shifted-moment formula are absent from the record; this assessment is retained as supplied curation, not as an independently verified theorem or status determination.

Short English statement

The proposed recipe predicts a smoothed average of quadratic-character Dirichlet polynomials from an integral of swap terms, with error $O(D^{1-\delta})$. The supplied record omits the formulas and hypotheses needed to make this precise. Its dated curation labels the problem partially solved and reports a conditional result for a length exponent below $2$; these claims require independent mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004289$, title "Long Dirichlet-Polynomial Recipe for Quadratic $L$-Functions," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2019$ and names no proposer.
• Supplied original report citation: Analytic Number Theory (2020), Oberwolfach Reports, pp. 3141--3205, DOI 10.4171/owr/2019/50. https://doi.org/10.4171/owr/2019/50. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. The relevant problem passage and full recipe are not reproduced in the record.
• Supplied literature reference: Conrey--Rodgers, "quadratic twists of long Dirichlet polynomials." https://doi.org/10.1007/s00209-025-03848-x.
• Supplied literature reference: Conrey--Rodgers, "conditional theorem." https://arxiv.org/abs/2208.01783.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It labels the problem partially solved, reports a conditional range with polynomial-length exponent below $2$ corresponding to the $0$- and $1$-swap terms, and describes the longer range and full shifted-moment prediction as remaining open. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.