Related papers: On Exponential Sums, Nowton identities and Dickson…

This paper derives a way to express differentiable complex-valued functions as the sum of powers of $(1-e^{\lambda x})$, where $\lambda\in\mathbb{R}$, with an explicit formula for the remainder. This formulation is then used to associate an…

We consider the linear vector space formed by the elements of the finite fields $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let $\{a_1,\ldots,a_r\}$ be a basis of this space. Then the elements $x$ of $\mathbb{F}_q$ have a unique…

Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) =…

Let $\Gamma$ be a cocompact, oriented Fuchsian group which is not on an explicit finite list of possible exceptions and $q$ a sufficiently large prime power not divisible by the order of any non-trivial torsion element of $\Gamma$. Then…

This paper presents a family of rapidly convergent summation formulas for various finite sums of the form $\sum_{k=0}^{\lfloor x\rfloor}f(k)$, where $x$ is a positive real number.

For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $\Phi\left(n\right)$ denote, respectively, the number of subsets $A$ of…

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…

We consider both finite and infinite power chi expansions of $f$-divergences derived from Taylor's expansions of smooth generators, and elaborate on cases where these expansions yield closed-form formula, bounded approximations, or analytic…

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the…

There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called…

Given an elliptic curve C, we study here $N_k = #C(F_{q^k})$, the number of points of C over the finite field F_{q^k}. This sequence of numbers, as k runs over positive integers, has numerous remarkable properties of a combinatorial flavor…

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example,…

Let $f(n)$ be an arithmetic function with $f(n) \ll n^\alpha$ for some $\alpha\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left(…

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…

We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in…

By using a variant of Kowalski's large sieve for Frobenius in compatible systems, we obtain zero-density estimates for arguments of $\ell$-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic…

We give nontrivial bounds for the bilinear sums $$ \sum_{u = 1}^{U} \sum_{v=1}^V \alpha_u \beta_v \mathbf{\,e}_p(u/f(v)) $$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$…

Let $F({\bf x})={\bf x}^tQ_{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x} =(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf x}\in\mathbb{R}_{\ge…

Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, \chi$ be a non-principal Dirichlet character modulo $q$. Let $\varepsilon$ be a sufficiently small positive constant, $A$…

Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…