Limiting Distributions of Mixed Character Sums

Definitions and mathematical statement

Write $e(u)=\exp(2\pi i u)$, using the standard analytic number theory convention, which the supplied record does not explicitly define. For each prime $q$, let $\chi_q$ denote the deterministic character referred to in the source. The surrounding excerpt concerns Dirichlet characters, apparently modulo $q$, but the prescription selecting $\chi_q$ is missing. The supplied literature assessment calls it a quadratic character; the main statement merely calls it a prescribed deterministic character. These descriptions require reconciliation before fixing the character family.

Let $\Theta$ have the uniform probability distribution on $[0,1]$. For a real cutoff $x$, interpret the displayed sum in the usual positive-integer sense as the complex-valued random variable \[ S(\chi_q,x,\Theta)=\sum_{\substack{n\in{\mathbb{Z}}\\1\leq n\leq x}}\chi_q(n)e(n\Theta). \] For every Borel set $B\subseteq{\mathbb{C}}$, its law is \[ \mu_{q,x}(B)=\int_0^1\mathbf{1}_{B}\!\left(\sum_{\substack{n\in{\mathbb{Z}}\\1\leq n\leq x}}\chi_q(n)e(n\theta)\right)\,\mathrm d\theta. \] The requested problem is to determine the limiting distribution of this sum as $q\to\infty$ through primes, with randomness coming from $\Theta$. In particular, after specifying the cutoff regime, the unnormalized formulation would ask for a probability measure $\mu$ on ${\mathbb{C}}$ such that \[ \mu_{q,x(q)}\Longrightarrow\mu, \] where $x(q)$ is only a placeholder for the cutoff regime that must be supplied; it is not an additional hypothesis extracted from the record. A fixed cutoff would instead mean $\mu_{q,x}\Longrightarrow\mu$ for that fixed $x$. The record does not say whether $x$ is fixed, depends on $q$, or parametrizes a stochastic process. It also supplies no centering or normalization for the sum in the question. Thus it does not determine a unique precise limiting-distribution problem.

The damaged original excerpt also mentions functions $F_{k,\chi,\alpha,\beta}(t)$, integers $k\in\{0,\ldots,q-1\}$, the strict summation boundaries $\alpha q<n<(\alpha+\beta)q$, and summands $\chi(n)e(n(k+t)/q)$. Its prefactor appears to involve $e(-\alpha k)$ and $q^{1/2}$, but the extracted formula is not reliably delimited. The domains of $\alpha,\beta,t$ and the connection of these functions to the requested limit are unspecified. This auxiliary excerpt therefore does not justify imposing its apparent scaling or a process interpretation on the stated question.

Short English statement

Determine the limiting probability law of a deterministic character sum when its phase is uniform on $[0,1]$ and the prime modulus tends to infinity. The supplied record leaves the character prescription, cutoff regime, and any intended normalization or process interpretation unresolved, so the target requires clarification before a unique precise formulation is possible.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30006452$, source problem code catalogue entry, title Limiting Distributions of Mixed Character Sums. No proposer is named; the supplied proposed year is 2025.
• Supplied source citation: Analytic Number Theory (2026), Oberwolfach Reports, pp. 2701--2764, DOI 10.4171/owr/2025/51. https://doi.org/10.4171/owr/2025/51. The background attributes the problem to workshop open-problem discussions collected in Oberwolfach Reports.
• Supplied literature source: Oberwolfach Report 51/2025. https://doi.org/10.4171/owr/2025/51.
• Supplied literature source: Distribution of mixed character sums. https://arxiv.org/abs/2510.06161.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the cited work proves limiting process laws for shifted mixed character sums with characters sampled from families, but does not determine the stated limit for a deterministic quadratic character with uniform phase. This is supplied dated literature triage, not an independently verified assessment; its time-sensitive status and claims require independent checking.
• The original excerpt refers to references [1] and [2], without supplying their bibliographic identities or an explicit correspondence to the links above. The registry labels the statement corrected_verified and describes the changes as repairs to formula delimiters, notation, grammar, and typography with mathematical content and scope preserved. These are supplied editorial claims, not independent mathematical verification.