arXiv subjects

Yixiu Xiao

Publications and source records attributed to Yixiu Xiao.

6 recordsLinked to original sources

arXiv2026

Kloosterman sign changes with moduli having at most five prime factors

On square-free moduli $q\in(X,2X]$ having at most five prime factors, we prove that each sign of the normalized Kloosterman sum $\operatorname{Kl}(1;q)$ occurs $\gg X/\log X$ times. This improves the recent unconditional result of Zhang and Zhong for moduli with at most six prime factors. Building on their analytic estimates and optimized Selberg sieve, we replace their truncated divisor penalty by a geometric half-weight. The new weight retains the $P_5$ exclusion threshold and is a positive linear combination of two standard two-parameter truncated divisor weights, so the Zhang--Zhong transference argument applies without alteration. After transference, the relevant pointwise coefficient is reduced from $5/16$ to $5/32$, which yields a positive final sieve margin.

math.NT↗

arXiv2026

Moment Estimates and Discrepancy for Sums of Square Roots Modulo One

Let $k\ge 2$ be fixed. We study the distribution modulo one of the $n^k$ sums \begin{equation*} \sqrt{a_1} + \cdots + \sqrt{a_k}, \qquad 1\le a_1, \dots, a_k \le n, \end{equation*} counted with multiplicity. For \begin{equation*} S(h,n) = \sum_{n/2\le a\le n} \mathbf{e}(h\sqrt{a}), \qquad \mathbf{e}(x) = \exp(2\pi i x), \end{equation*} we prove second- and fourth-moment estimates matching the diagonal scale up to a factor $n^\varepsilon$. More precisely, \begin{equation*} \sum_{H/2\le h\le H} \left| S(h,n) \right|^2 \ll_{\varepsilon,\delta} Hn^{1+\varepsilon} \end{equation*} uniformly for $H\ge n^{1/2+\delta}$, and \begin{equation*} \sum_{H/2\le h\le H} \left| S(h,n) \right|^4 \ll_{\varepsilon,\delta} Hn^{2+\varepsilon} \end{equation*} uniformly for $n^{1/2+\delta} \le H \le n^{2/3}$, where $0<\delta<1/6$ in the fourth-moment estimate. Combining the second-moment bound with pointwise exponential-sum estimates and the Erd\H{o}s--Tur\'an inequality, we obtain \begin{equation*} D_k(n) \le n^{-\rho_k+o(1)}, \qquad \rho_k = \frac{71k+26}{26k+116}, \end{equation*} as $n\to\infty$, where $D_k(n)$ denotes the discrepancy with respect to arbitrary subintervals of $[0,1)$.

math.NT↗

arXiv2026

Large sieve inequality for sums of Legendre symbols over short intervals

Using the Burgess bound and the Selberg sieve, we obtain an upper bound for the second moment of sums of Legendre symbols over intervals , with the modulus ranging over primes . The bound is nontrivial and yields a power saving in , uniformly for , provided that , where as . This may be viewed as a short-interval analogue of a result of D. R. Heath-Brown (1995) on moments of quadratic character sums over the initial interval . In particular, it implies that, for any prescribed interval of this length, the quadratic residues and non-residues are asymptotically equidistributed for almost all primes . We also establish estimates for higher moments conditionally on the Generalised Riemann Hypothesis. These bounds rely on a sharp uniform estimate for the number of tuples of integers in a shifted interval whose product is a square.

math.NT↗

arXiv2026

Least zero of pairs of additive cubic equations

An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms \[ \begin{cases} F = c_{1}x_1^3 + c_{2}x_2^3 + \cdots + c_{n}x_n^3 = 0, \\ G = d_{1}x_1^3 + d_{2}x_2^3 + \cdots + d_{n}x_n^3 = 0, \end{cases} \] under the "$M$-good" condition for $n \ge 16$, where $c_{1}, \dots, c_{n}$ and $d_{1}, \dots, d_{n}$ are integers. Additionally, a range is derived for the probability that randomly selected simultaneous equations satisfy the $M$-good condition.

math.NT↗

arXiv2026

Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes

We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Salié sums modulo a large prime $q$. We use these bounds to study, for fixed integers $a,b\not \equiv 0 \bmod q$, the distribution ofsolutions to the congruence $x^2 \equiv ap+b \bmod q$, over primes $p\le P$. This is similar to the recently studied case of $b = 0$, however the case $b\not \equiv 0 \bmod q$ exhibits some new difficulties.

math.NT↗

arXiv2024

Least zero of a cubic form

An explicit upper bound is established for the least non-trivial integer zero of an arbitrary cubic form $C \in \mathbb{Z}[X_1,...,X_n],$ provided that $n \geq 14.$

math.NT↗