Denote by $\mathrm{Kl}(a,n)$ the normalized Kloosterman sum modulo $n$. In this paper, we study the moments of $|\mathrm{Kl}(1,n)|$ on average over $n\leqslant X$, and show that \begin{align*} \sum_{n\leqslant X}|\mathrm{Kl}(1,n)|\asymp X(\log X)^{\frac{8}{3\pi}-1}. \end{align*} The lower bound is proven by considering a subfamily of moduli which factorize in a nice way such that vertical Sato--Tate distributions of Kloosterman sums (as variants of Katz) apply. The upper bound is based on a new density version of upper bound sieves, combining with averages of divisor functions in arithmetic progressions and a similar vertical Sato--Tate distribution.
We extend the Nair--Tenenbaum estimates for short sums of nonnegative arithmetic functions at polynomial values to mixed moments involving Kloosterman sums. Combining these estimates with lower bounds obtained by refining the method of Fouvry and Michel, we prove that, for every fixed nonzero integer \(a\) and fixed \(...
Majorization is a fundamental tool for comparing how"spread out"the entries of two vectors are. Key majorization results were obtained for the integers by Hardy, Littlewood and P\'olya and for $\mathbb{Z}_p$ by Lev. In this paper, we establish a powerful majorization theorem in $\mathbb{Z}_2^n$ that is an analogue of L...
Sophie Huczynska, F. Rakhmonov, Chi-Hoi Yip· 0 citations
We study the distinctness and algebraic degrees of inverted Kloosterman sums over finite extensions of $\mathbb{F}_p$. For each $d\geq 2$, we prove that the $(d-1)$-dimensional sums over $\mathbb{F}_{p^2}$ are equal precisely when their parameters are Frobenius conjugate, provided that $p\equiv 1\pmod d$, $p>2d+1$, and...
We give a randomized algorithm that, for an $m\times n$ matrix whose columns have Euclidean norm at most one, finds a signing with discrepancy below $99$ in expected $\widetilde O(L_{\mathrm{in}}+n^\omega)$ bit operations, where $L_{\mathrm{in}}$ is the input encoding length and $\omega$ is the exponent of matrix multi...
As a natural generalization of the classical hyper-Kloosterman family studied by D. Haessig and S. Sperber, we study the $k$-th symmetric power $L$-functions attached to a weighted hyper-Kloosterman family $$Kl_{n,m}(t;x_{1},\cdot\cdot\cdot,x_{n})=x_{1}^{m}+x_{2}\cdot\cdot\cdot+x_{n}+\frac{t}{x_{1}x_{2}\cdot\cdot\cdot...
We bound the error term in the counting function of cubic extensions of $\mathbb{Q}$ by $\mathcal{O}_\epsilon(X^{\theta+\epsilon})$, with $\theta<2/3$, improving upon results due to Bhargava, Taniguchi, and Thorne. The proof proceeds by relating a certain discriminant-reducing sieve, due to Bhargava, Shankar, and Tsime...
Victor Ahlquist· 0 citations
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