Uniform Zero-Distribution Limits for Characteristic Test Functions

Definitions and mathematical statement

For $y\geq0$, let \[ h_y(x)=\mathbf1_{[0,y]}(x)= \begin{cases}1,&0\leq x\leq y,\\0,&x\notin[0,y].\end{cases} \] The source conjectures that its zero-distribution limit formula for the Riemann zeta function or Dirichlet $L$-functions remains valid for these characteristic test functions, uniformly in $y$. The formula itself is not contained in the supplied record. In particular, the record does not identify whether the statistic concerns individual zeros, gaps, pairs of zeros or another quantity, nor its normalization or limiting expression.

To record the uniformity requirement without inventing that formula, denote the missing zero statistic by $A_T(h)$, its proposed limiting functional by $A(h)$, and the intended set of interval endpoints by $Y$. With this notation only, the requested extension would have the form \[ \lim_{T\to\infty}\sup_{y\in Y} \left|A_T(h_y)-A(h_y)\right|=0. \] Here $T$ is a placeholder for the absent asymptotic parameter, and $Y$, $A_T$ and $A$ must be taken from the original formula. If its normalization places an explicit main term elsewhere, the corresponding uniform error statement must be used instead. The display specifies what uniform convergence would mean; it does not assert an independently defined zero-distribution theorem. In a measure formulation, if applicable, $A_T(h_y)=\mu_T([0,y])$ and $A(h_y)=\mu([0,y])$; the record does not supply such measures or guarantee that they are probability measures.

The distinction between every fixed $y$ and uniformity over the full intended $Y$ is retained. The interval is the closed interval $[0,y]$, including its endpoints; no continuity, absence of atoms or boundary convention is supplied to justify replacing it by another interval. It is also unspecified whether uniformity is over all $y\geq0$, bounded intervals of $y$, or a growing parameter range. For Dirichlet $L$-functions the record gives no character, primitivity, conductor or averaging conditions, and no Riemann-hypothesis assumption is added.

The supplied literature assessment, checked 2026-08-21, says the report explicitly labels this extension a conjecture and that located partial distribution results did not establish the full uniform statement. The verification field says the conjecture label was removed when restoring the notation; this editorial removal does not supply a proof. The open label remains dated source metadata.

Short English statement

The requested extension replaces the original test function by the indicator of $[0,y]$ and requires the error to tend to zero uniformly in the intended range of $y$. The underlying zero statistic, limiting formula and parameter ranges are missing, so they must be recovered before this becomes a fully specified analytic assertion.

Sources

• Local UnsolvedMath registry, permanent ID 30000135, source code catalogue entry, "Uniform Zero-Distribution Limits for Characteristic Test Functions," including original wording, cleaned wording, verification note and dated literature assessment.
• Theory of the Riemann Zeta and Allied Functions (2005), Oberwolfach Reports, pp. 2419--2490, DOI 10.4171/owr/2004/46: https://doi.org/10.4171/owr/2004/46. This is also the sole supplied literature reference, labelled "original report." The report citation uses 2005, whereas the proposed-year field and DOI use 2004.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external reference has been visited or independently assessed for this draft.