Related papers: On the support of the Kloosterman paths

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums, as their parameter varies modulo a prime tending to infinity. Using independence of Kloosterman sheaves, we prove convergence in the…

We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power p^n of a fixed odd prime p, a pure depth-aspect analogue of theorems of Kowalski-Sawin and…

In this paper, we consider the distribution of the continuous paths of Dirichlet character sums modulo prime $q$ on the complex plane. We also find a limiting distribution as $q \rightarrow \infty$ using Steinhaus random multiplicative…

In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.

We give optimal bounds for matrix Kloosterman sums modulo prime powers extending earlier work of the first two authors on the case of prime moduli. These exponential sums arise in the theory of the horocyclic flow on $\mathrm{GL}_n$.

For $q$ prime, $X \geq 1$ and coprime $u,v \in \mathbb{N}$ we estimate the sums \begin{equation*} \sum_{\substack{p \leq X \substack p \equiv u \hspace{-0.25cm} \mod{v} p \text{ prime}}} \text{Kl}_2(p;q), \end{equation*} where…

We prove two estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker.

We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2\pi i (x \lfloor f(n)\rfloor+ y \lfloor f(n)\rfloor^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit…

For a prime $p$, we consider Kloosterman sums $$ K_{p}(a) = \sum_{x\in \F_p^*} \exp(2 \pi i (x + ax^{-1})/p), \qquad a \in \F_p^*, $$ over a finite field of $p$ elements. It is well known that due to results of Deligne, Katz and Sarnak, the…

We prove power-saving bounds for general Kloosterman sums on $\operatorname{Sp}(4)$ associated to all Weyl elements via a stratification argument coupled with $p$-adic stationary phase methods. We relate these Kloosterman sums to the…

We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to…

Emmanuel Kowalski and William Sawin proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S(a,b0;p)/p^{1/2} converge in the sense of…

We investigate the distribution of modular inverses modulo positive integers $c$ in a large interval. We provide upper and lower bounds for their box, ball and isotropic discrepancy, thereby exhibiting some deviations from random point…

G. Ricotta and E. Royer (2018) have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums $S(a,b;p^n)/p^(n/2) converge in law in the Banach space of complex-valued continuous function…

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$-periodic complex valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this…

We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight…

We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums.…

We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to composite modulo $q$, that is, for the exponential sum of the type \[ \sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2\pi…

We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum…

Let $p$ be a prime number and $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol modulo $p$. The \emph{Legendre path} attached to $p$ is the polygonal path whose vertices are the normalized character sums $\frac{1}{\sqrt{p}} \sum_{n\leq…