Author

Yi-Xiu Xiao

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Preprint Sep 2026

Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors

Let $\Kl(1;q)$ denote the normalized Kloosterman sum modulo $q$. We prove that, for each sign, there are $\gg X/\log X$ square-free moduli $q\in(X,2X]$ with at most four prime factors for which $\Kl(1;q)$ has that sign. The proof combines the Selberg sieve with equidistribution and mean-square estimates for K...

Yi-Xiu Xiao, Hong-Ze Li · 0 citations

Preprint Sep 2026

Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights

We prove an asymptotic formula for correlations of divisor sums with quadratic cutoff functions shifted independently by $s,u\in[0,1/2]$. The error is $O_g(Y/(\log Y)^2)$, uniformly for $Y^{1/5}\leq R\leq Y^{1/3}$, including coincident shifts. The main term is an explicit bilinear form in the first and second der...

Yi-Xiu Xiao, Hong-Ze Li · 1 citation · ⚡1

Preprint Sep 2026

Arithmetic structure of $L_2$-norms of ${\mathrm{SL}}_2(\mathbb{Z})$ matrices

For a matrix $\gamma\in\mathrm{SL}_2({\mathbb Z})$, we define \[ {\mathcal R}(\gamma)=a_1^2+a_2^2+a_3^2+a_4^2, \qquad \text{where} \ \gamma=\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}, \] and let $S_{\mathrm{sq}}(X)$ count the number of matrices $\gamma$ with $\|\gamma\|_\infty = \max\{|a_1|, |a_2|,|a_3|,|a_4|\} \leq...

Igor E. Shparlinski, Yi-Xiu Xiao · 0 citations