Lower Bounds for Twisted-Multiplicative Exponential Sums

Definitions and mathematical statement

For a real number $t$, write $e(t)=\exp(2\pi i t)$. The supplied formulation uses the normalized complete exponential sum \[ W_f(a;q)=\frac{1}{\sqrt q}\sum_{u\bmod q}e\!\left(\frac{a f(u)}q\right). \] Here $q$ is a positive integer modulus. This formula requires a specified interpretation of $f(u)$ modulo $q$, with the summand independent of the chosen representative of $u$. For example, an integer polynomial and an integer $a$ give such an interpretation. This example does not specify the intended admissible class: the supplied record does not state the hypotheses on $f$, the domain or quantification of $a$, or restrictions on the moduli. If rational functions are intended, the treatment of poles and denominators also needs to be specified.

For a real cutoff $X\geq1$, the quantity singled out by the supplied reconstructed statement is \[ S_{f,a}(X)=\sum_{\substack{q\in{\mathbb{N}}\\1\leq q\leq X}}|W_f(a;q)|. \] Requested lower-bound problem. Obtain lower bounds for $S_{f,a}(X)$ of the same order as the best available upper estimates, under the intended hypotheses on $f$, $a$, and $q$. The record supplies neither those hypotheses nor the upper-bound expression to be matched, its parameter dependence, or its uniformity range. Consequently this is an explicitly incomplete formulation of the requested comparison, rather than a fully quantified conjectural inequality.

The original extract asks for lower bounds for "the same quantities (with modulus)", whereas its adjacent Problem 1 concerns sums of twisted-multiplicative functions without modulus. In the reconstructed statement, "with modulus" is interpreted as taking the complex absolute value $|W_f(a;q)|$. The antecedent quantities, the referenced bound (2), and Theorem 1 are absent, so the scope of the original plural request cannot be recovered from this record alone. No twisted-multiplicativity identity is supplied.

The accompanying source remark attributes estimates of related sums to Hooley [1964] for Kloosterman sums, formally corresponding to $f(T)=T+T^{-1}$, and to Fouvry and Michel [2003] for "generic" rational functions. It reports an upper bound of size $X(\log\log X)^{c_f}$, with $c_f>0$, for the missing bound (2). It also describes Theorem 1 as giving the qualitatively new conclusion that the average of $W_f(a;q)$ tends to zero. These are supplied contextual claims with missing hypotheses; the displayed historical bound is not identified here as the upper estimate to be matched, and cancellation without absolute values does not specify the requested absolute-value lower bound.

Short English statement

The supplied problem asks for lower bounds matching upper estimates for the sum of the absolute values of normalized exponential sums over moduli. The admissible functions, parameter ranges, modulus restrictions, and target upper estimate are missing, so a fully quantified conjecture requires source clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30005273$, title "Lower Bounds for Twisted-Multiplicative Exponential Sums", and exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and supplies no named proposer.
• 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. The record gives proposed year 2022. Its extracted passage includes a page marker 2931, repeated material, and an incomplete discussion surrounding Problems 1 and 2.
• Supplied literature source: Kowalski and Soundararajan. https://doi.org/10.1007/978-3-031-05331-3_13. No fuller bibliographic description is supplied.
• Supplied literature source: Oberwolfach Report 50. https://ems.press/content/serial-article-files/46986.
• Historical references appearing in the supplied extract: Hooley [1964], concerning Kloosterman sums, and Fouvry and Michel [2003], concerning "generic" rational functions. Full bibliographic entries and the relevant hypotheses are not supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that no general lower bounds matching the extracted twisted-multiplicative upper-bound problem are established in the cited primary literature. This is supplied dated curation derived from the OWR collection, not independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The registry labels the reconstructed statement corrected_verified and says it was checked against the official report. This is a supplied verification claim; no independent source inspection or mathematical verification is asserted here.