Related papers: Estimates for Character Sums with Various Convolut…
For a set $A$ of non-negative integers, let $R_A(n)$ denote the number of solutions to the equation $n=a+a'$ with $a$, $a'\in A$. Denote by $\chi_A(n)$ the characteristic function of $A$. Let $b_n>0$ be a sequence satisfying $\limsup_{n\to…
We extend the results obtained by E. Ntienjem to all positive integers. Let $\EuFrak{N}$ be the subset of $\mathbb{N}$ consisting of $\,2^{\nu}\mho$, where $\nu$ is in $\{0,1,2,3\}$ and $\mho$ is a squarefree finite product of distinct odd…
We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime $p$.
The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…
A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq…
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…
Given a Dirichlet character $\chi$ modulo $q$ and its associated $L$-function, $L(s,\chi)$, we provide an explicit version of Burgess' estimate for $|L(s, \chi)|$. We use partial summation to provide bounds along the vertical lines $\Re{s}…
We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.
We consider Weil sums of binomials of the form $W_{F,d}(a)=\sum_{x \in F} \psi(x^d-a x)$, where $F$ is a finite field, $\psi\colon F\to {\mathbb C}$ is the canonical additive character, $\gcd(d,|F^\times|)=1$, and $a \in F^\times$. If we…
We revisit the proposed equality between discrete Fourier transforms of $p$-adic $\Gamma_p$--values and $p$-adic $L$--derivatives for odd characters modulo a prime $p$. The clean identity is false in general. Building on Coleman reciprocity…
In this article, we obtain certain estimate for the shifted convolution sum involving the Fourier coefficients of half-integral weight cusp forms.
For all natural numbers $n$, we discuss the evaluation of the convolution sum, $\underset{\substack{{(l,m) \in \mathbb{N}_0^2} \\ {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)$, where $\alpha\beta=14,22,26$. We generalize the…
We prove an estimate for multi-variable multiplicative character sums over affine subspaces of $\mathbb A^n_k$, which generalize the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character…
In this paper, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)\chi(n)$. We prove a new Omega result in the region $\exp((\log q)^{\frac12+\delta})\le N\le\sqrt q$, where $q$ is the…
In this article, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)\chi(n)$. We prove an Omega result in the region $\exp((\log q)^{\frac12+\varepsilon})\le N\le\sqrt q$, where $q$ is…
In this paper, we provide formulas for partial sums of weighted averages over regular integers modulo $n$ of the $\gcd$-sum function with any arithmetic function. Many interesting applications of the results are also given.
Let $\chi=\chi_q$ be a primitive character mod $q$ and fix $\Delta>0$. In 1989 Graham and Ringrose gave strong bounds on character sums $\sum_{M<n\leq M+N} \chi(n)$ in intervals of length $N=q^\Delta$ whenever $q$ is squarefree and is…
In this paper we obtain a new fully explicit constant for the P\'olya-Vinogradov inequality for primitive characters. Given a primitive character $\chi$ modulo $q$, we prove the following upper bound \begin{align*} \left| \sum_{1 \le n\le…
We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive…
We provide a systematic method to compute arithmetic sums including some previously computed by Alaca, Alaca, Besge, Cheng, Glaisher, Huard, Lahiri, Lemire, Melfi, Ou, Ramanujan, Spearman and Williams. Our method is based on quasimodular…