Keating–Snaith Distribution of Central L-Values

Definitions and mathematical statement

The supplied record attributes the following value-distribution conjecture to Keating--Snaith. Its family specifications and logarithm conventions are incomplete, so the formulation below is a parameterized statement requiring those choices to be resolved.

For a member $F$ of an $L$-function family, write $L_c(F)$ for its central value, in a normalization that remains to be specified. Let $\lambda(F)$ denote the intended real-valued logarithmic statistic of $L_c(F)$. For positive central values this can mean $\log L_c(F)$; complex, negative, or zero central values require additional conventions not supplied in the record.

The three proposed families and normalization parameters are \[ \begin{array}{c|c|c|c} j & \text{family description} & \mu_j(T) & \sigma_j(T)^2\\ \hline 1 & \text{primitive Dirichlet characters modulo }q,\quad T=q & 0 & \tfrac12\log\log q\\ 2 & \text{quadratic characters }\chi_d,\quad T=X & \tfrac12\log\log X & \log\log X\\ 3 & \text{quadratic twists }f\otimes\chi_d\text{ with root number }+1,\quad T=X & -\tfrac12\log\log X & \log\log X \end{array} \] Here a root number is the sign or unit-modulus factor in the functional equation; the third row retains the source's restriction that this factor equals $+1$. The hypotheses on $f$, the discriminants or other indices represented by $d$, their range relative to $X$, and the central-point normalization are not specified by the supplied record.

For each $j\in\{1,2,3\}$, suppose that a finite nonempty family $\mathcal F_j(T)$, a probability measure $\mathbb P_{j,T}$ on that family, and the statistic $\lambda$ have been specified. The intended asymptotic Gaussian assertion can then be expressed as follows: for every real $t$, \[ \lim_{T\to\infty} \mathbb P_{j,T}\!\left( \frac{\lambda(F)-\mu_j(T)}{\sigma_j(T)}\le t \right) =\Phi(t), \qquad \Phi(t)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{t}e^{-u^2/2}\,{\,\mathrm{d}} u. \] The limit must be taken along the admissible parameter values of the chosen family. Uniform counting would give $\mathbb P_{j,T}(A)=\#A/\#\mathcal F_j(T)$, but the record does not explicitly specify the sampling measure.

The displayed $\mu_j$ and $\sigma_j^2$ preserve the source's proposed Gaussian centering and variance scales. The phrase "mean asymptotic to $0$" in the first family has no ordinary ratio-asymptotic meaning. Moreover, convergence in distribution alone does not assert convergence of empirical means or variances. The source does not determine whether separate moment asymptotics are intended.

In particular, the first family generally has complex central values, so a real Gaussian statement requires clarification of whether $\log|L_c(F)|$ is intended. In every family, vanishing central values require a convention or an explicitly defined restriction. The root-number-$+1$ condition in the third family does not itself guarantee nonvanishing. These unresolved choices prevent this parameterized target from being a fully specified conjecture.

Short English statement

The supplied Keating--Snaith conjecture predicts Gaussian fluctuations of logarithmic central $L$-values in three families, with the centering and variance scales displayed above. The family ranges, sampling measures, hypotheses on $f$, and treatment of complex or vanishing values need clarification. The supplied "partially solved" label is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: permanent ID $30003580$, title "Keating--Snaith Distribution of Central L-Values," exact source problem code catalogue entry. Registry references: data/unsolvedmath/problems.json and catalog/problems.json; this draft uses the supplied complete record without independently inspecting those files.
• Supplied report citation: Automorphic Forms and Arithmetic (2018), Oberwolfach Reports, pp. 2475--2537, DOI 10.4171/owr/2017/40. https://doi.org/10.4171/owr/2017/40. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The record gives proposed year 2017 and attributes the conjecture to Keating--Snaith; no individual proposer is named.
• Supplied literature source, labeled "Radziwill--Soundararajan": https://arxiv.org/abs/1403.7067.
• Supplied literature source, labeled "Conditional lower bounds": https://doi.org/10.4064/aa230805-3-1.
• Supplied literature source, labeled "OWR report": https://doi.org/10.4171/owr/2017/40.
• Supplied literature assessment, checked on 2026-08-21: status "partially solved." The supplied assessment says that, for quadratic twists, sharp upper bounds and a one-sided central limit theorem have been proved, that conditional lower bounds were subsequently obtained, and that the full conjectural value distribution remains unestablished. This is supplied dated curation derived from the OWR material, not independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and describes its editing as repairs to notation, typography, and sentence structure while preserving mathematical content. That supplied label does not constitute independent mathematical verification of this draft.