Related papers: Character sums to prime power moduli evaluated at …

Let $q$ be a prime power and $r$ a positive even integer. Let $\mathbb{F}_{q}$ be the finite field with $q$ elements and $\mathbb{F}_{q^r}$ be its extension field of degree $r$. Let $\chi$ be a nontrivial multiplicative character of…

We give an estimate of exponential sums over singular binary quintic forms in a characteristic-free form, based on the Waring decomposition of binary forms. This extends the method on our preceding result on the space of binary quartics to…

We obtain explicit estimates for the mixed character sum $S= S(\chi,g,f,p^m) = \sum_{x=1}^{p^m} \chi (g(x)) e_{p^m}(f(x))$, where $p^m$ is a prime power, $\chi$ is a multiplicative character mod $p^m$ and $f,g$ are rational functions over…

We propose a new frame work of $p$-adic neural network. Unlike the original $p$-adic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi using a family of characteristic functions indexed by hyperparameters of precision as…

We study large values of quadratic character sums with summation lengths exceeding the square root of the modulus. Assuming the Generalized Riemann Hypothesis, we obtain a new Omega result.

The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on…

We propose higher-order generalizations of Jacobsthal's $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\binom{ip}{p}$ ($i=1,2,\dots$) that are…

We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2\pi iag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning…

For a normalized newform $g \in S_{k}(\Gamma_{0}(N))$ with complex multiplication by an imaginary quadratic field $K$, there is a mock modular form $F^{+}$ corresponding to $g$. K. Bringmann et al. modified $F^{+}$ in order to obtain a…

In this paper we consider the problem of estimating character sums to composite modulus and obtain some progress towards removing the cubefree restriction in the Burgess bound. Our approach is to estimate high order moments of character…

In their recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the…

Various algebraic properties of Heilbronn's exponential sum can be deduced through the use of supercharacter theory, a novel extension of classical character theory due to Diaconis-Isaacs and Andre. This perspective yields a variety of…

We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in common with analogous formulas proved previously for other vector…

We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan on quadratic Dirichlet…

We establish a new bound for short character sums in finite fields, particularly over two-dimensional grids in $\mathbb{F}_{p^3}$ and higher-dimensional lattices in $\mathbb{F}_{p^d}$, extending an earlier work of Mei-Chu Chang on Burgess…

We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli $c$, using Fourier analysis on $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ and an amplification argument with non-abelian characters. For sums…

We obtain new bounds on some trilinear and quadrilinear character sums, which are non-trivial starting from very short ranges of the variables. An application to an apparently new problem on oscillations of characters on differences between…

We improve a recent result of B. Hanson (2015) on multiplicative character sums with expressions of the type $a + b +cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a + b(c+d)$.…

We shall introduce and study certain truncated sums of Hecke eigenvalues of $GL_2$-automorphic forms along quadratic polynomials. A power saving estimate is established and new applications to moments of critical $L$-values associated to…

Hybrid character sums are an important class of exponential sums which have nice applications in coding theory and sequence design. Let $\gf_{p^m}$ be the finite field with $p^m$ elements for a prime $p$ and a positive integer $m$. Let…