9 papers · 1 filter
Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors
Yixiu Xiao, Hongze Li
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 pr…
Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights
Yixiu Xiao, Hongze Li
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 \(O_g(Y/(\log Y)^2)\), unif…
Arithmetic structure of -norms of matrices
Igor E. Shparlinski, Yixiu Xiao
For a matrix , we define \[ {\mathcal R}(γ)=a_1^2+a_2^2+a_3^2+a_4^2, \qquad \text{where} \ γ=\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}, \] and l…
Kloosterman sign changes with moduli having at most five prime factors
Yixiu Xiao
On square-free moduli having at most five prime factors, we prove that each sign of the normalized Kloosterman sum occurs times…
Moment Estimates and Discrepancy for Sums of Square Roots Modulo One
Yixiu Xiao
Let be fixed. We study the distribution modulo one of the sums \begin{equation*} \sqrt{a_1} + \cdots + \sqrt{a_k}, \qquad 1\le a_1, \dots, a_k \le n, \end{equation*}…
Large sieve inequality for sums of Legendre symbols over short intervals
Marc Munsch, Igor Shparlinski, Yu-Chen Sun +1
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 . Th…
Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes
Igor E. Shparlinski, Yixiu Xiao
We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Salié sums modulo a large prime . We use these bounds to study, for fixed integers $a,b\not \e…
Least zero of pairs of additive cubic equations
Yixiu Xiao, Hongze Li
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 =…
Least zero of a cubic form
Yixiu Xiao, Hongze Li
An explicit upper bound is established for the least non-trivial integer zero of an arbitrary cubic form provided that