Cancellation in Twisted-Multiplicative Exponential Sums

Definitions and mathematical statement

Write $e(t)=\exp(2\pi i t)$ for real $t$. The supplied record introduces the normalized exponential sum \[ W_f(a;q)=\frac{1}{\sqrt q}\sum_{u\bmod q}e\!\left(\frac{a f(u)}q\right) \] and asks about \[ S_f(a;X)=\sum_{\substack{1\le q\le X\\q\in{\mathbb{Z}}}}W_f(a;q),\qquad X\ge1. \] These formulas are meaningful only after specifying the admissible functions $f$, the parameter $a$, and a convention making the summands well defined on residue classes. These hypotheses are absent from the supplied record. In particular, the record mentions rational functions, for which poles and denominators modulo $q$ require additional conventions; the displayed sum cannot automatically be interpreted as a sum over every residue class in that setting.

The main requested task is to develop methods for estimating $S_f(a;X)$ that retain cancellation between different moduli, without first replacing every $W_f(a;q)$ by $|W_f(a;q)|$. The original excerpt phrases this more generally as finding other methods for sums of twisted-multiplicative functions, without taking absolute values. No twisted-multiplicativity identity, admissible function class, quantitative target, or uniformity requirements are supplied, so this is a research objective rather than a fully quantified conjectural inequality.

The original excerpt also asks for lower bounds for the same quantities "with modulus." It does not specify whether this means a lower bound for $|S_f(a;X)|$, for $\sum_{q\le X}|W_f(a;q)|$, or for another quantity defined in the missing surrounding text. This additional request is retained as unresolved; no lower-bound inequality is asserted here.

The accompanying source fragment attributes earlier estimates to Hooley (1964) for Kloosterman sums, written there as $f(T)=T+T^{-1}$, and to Fouvry and Michel (2003) for "generic" rational functions. It says that an unspecified Theorem 1 gives vanishing average for $W_f(a;q)$, in contrast with earlier upper bounds. The theorem, its hypotheses, and the referenced bound (2) are not supplied, so that assertion cannot be made into an unconditional statement here.

Short English statement

Develop methods that capture cancellation when normalized exponential sums are summed over positive moduli. The original excerpt additionally requests lower bounds involving absolute values. The admissible functions, parameter restrictions, rational-function conventions, precise estimates, and intended lower-bound quantity remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005272$, title: Cancellation in Twisted-Multiplicative Exponential Sums; exact source code catalogue entry. No individual proposer is supplied. The proposed year is recorded as 2022.
• Supplied report citation: Analytic Number Theory (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/50, pp. 2895--2960. https://doi.org/10.4171/owr/2022/50. Oberwolfach Reports are described in the record as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Kowalski and Soundararajan. https://doi.org/10.1007/978-3-031-05331-3_13.
• Supplied literature source: Oberwolfach Report 50. https://ems.press/content/serial-article-files/46986.
• The original excerpt attributes earlier estimates to Hooley (1964) and generalizations to Fouvry and Michel (2003), with the qualifications stated above. Full bibliographic references for these works are not supplied.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not an independently verified assessment. The record labels the problem open and states that the cited theory gives upper bounds for twisted-multiplicative sums, while a general method without taking absolute values and the requested lower bounds are not known. The background expressly calls this dated literature triage derived from the supplied Oberwolfach curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim; the present draft has not independently visited or checked that source.