bibkey: cochranegranvillezheng2026smooth authors: Todd Cochrane, Andrew Granville, and Junren Zheng year: 2026 title: Mixed incomplete character sums of rational functions with smooth moduli doi: 10.48550/arXiv.2601.10927 url: https://arxiv.org/pdf/2601.10927v1 claim: Corollary 2 gives short character-sum cancellation for a specified smooth-modulus class, with an exceptional integer and a length threshold; the stated parameter uniformity does not reach logarithmic intervals for general moving moduli. strata_touched: [] license: citation-only triage: anchor

Smooth-modulus character sums and the actual cutoff obligation

The inspected source is arXiv:2601.10927v1, submitted 16 January 2026. Theorem 1 and the parameter-uniformity paragraph are on p.2; Corollary 2 is on p.3. This note checks those statements and their applicability, not the full proof. It treats this version as a preprint and claims no Lean verification or complete current survey.

The source’s actual class and hypotheses

The paper defines $\mathcal{N}(y)$ as the positive integers with at most one prime factor in $(y,y^2]$, with all other prime-power divisors at most $y$. Smoothness of the radical alone is not this condition.

Corollary 2 states that for each fixed $\delta ,\epsilon >0$ there are $\eta =\eta (\delta ,\epsilon )>0$ and an explicitly determinable positive integer $\mathcal{M}=\mathcal{M}(\delta )$ such that, if

$$y=q^{\delta },q\in \mathcal{N}(y),(q,\mathcal{M})=1,$$

then every nonprincipal character $\chi$ modulo $q$ and every interval $I$ of length $H$ with $q\ge H\ge y^{1+\epsilon }$ satisfy

$$\left|\sum _{n\in I}\chi (n)\right|\ll H^{1-\eta }.$$

The implicit constant depends on the fixed parameters. The character need not be primitive. Theorem 1 treats mixed sums of rational functions; its exceptional low-degree case explicitly uses the primitive conductor, rather than automatically giving a saving in the full modulus. No numerical value of $\eta$ or onset threshold is certified here.

The p.2 paragraph says that their developed proof permits $\delta \to 0$ only with $\delta \gg 1/\log \log q$. For such a choice the length requirement still has scale at least

$$H\ge \exp \left(c\frac{\log q}{\log \log q}\right)$$

for a positive constant in that parameter regime. At $q$ comparable to the target integer $N$, setting $H=\log N$ would instead require $\delta \le \log \log N/((1+\epsilon )\log q)$, below this stated uniformity range. This is a limitation of the cited application, not a lower bound for every actual character sum.

FIB interfaces still required

The FIB theory volume §§181,182,202 retains the actual conductor, induced zero-character primes and multiplier costs. It has no all-source result placing the actual moduli in $\mathcal{N}(q^{\delta })$ or proving their coprimality with $\mathcal{M}$. Its bound $q_D<4N/g^2$ alone supplies neither those conditions nor the needed length comparison.

Character-sum cancellation also needs a justified sieve or other bridge before it supplies the reciprocal-prime mass missing from the same actual integer. The Bourgain–Lindenstrauss weighted theorem already supplies one such established input in its stated restricted range. The present source should be reused if an actual FIB family meets its hypotheses; reproving its smooth-modulus estimates does not fill these family-specific obligations or settle general Robin and RH.