On sums of coefficients of polynomials related to the Borwein conjectures

The Ramanujan Journal - Recently, Li (Int J Number Theory, 2020) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture....     

算术数列中的素变数指数和问题

旨在应用初等方法研究指数和问题,给出了算术数列中素变数非线性指数和的一个上界估计.     

关于算术级数的幂次和

主要研究了ζ函数关于模 q剩余类部分和,不仅得出了一个重要的渐近公式,而且将Kubert恒等式推广到赫尔维茨ζ函数、欧拉双Γ函数和贝努利多项式上.     

Rankin-Selberg coefficients in large arithmetic progressions

Let (λ_f(n))_n≥1 be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form f.We prove that,for any fixed η> 0,under the Ramanujan-Petersson conjecture for GL₂ Maass forms,the Rankin-Selberg coefficients(λ_f(n)²)_n≥1 admit a level of distribution θ=2/5+1/260-η in arithmetic progressions.     

Exponential sums involving automorphic forms for GL(3) over arithmetic progressions

Let f be a Hecke-Maass cusp form for SL(3;\mathrm{\mathbb{Z}}) with Fourier coefficients A_f(m; n); and let \varphi (x) be a C^{\infty } -function supported on [1; 2] with derivatives bounded by {\varphi }^{(j)}(x)\ll j 1. We prove an asymptotic formula for the nonlinear exponential sum {\mathrm{\Sigma }}_{n\equiv l\mathrm{mod}q}{A}_f(m,n)\phi (n/X)e{\left(3\left(kn\right)\right)}^{1/3}/q, where e(z)={\mathrm{e}}^{\mathrm{2}\pi iz} and k\in {\mathrm{\mathbb{Z}}}^{+}.     

Greedy algorithm, arithmetic progressions, subset sums and divisibility

We consider a number of density problems for integer sequences with certain divisibility properties and sequences free of arithmetic progressions. Sequences of the latter type that are generated by a computer using modifications of the greedy algorithm are also provided.     

Parity of sums of partition numbers and squares in arithmetic progressions

In this paper, we prove new infinite families of congruences modulo 2 for broken 11-diamond partitions by using Hecke operators.     

Borwein's conjecture on average over arithmetic progressions

We provide asymptotic formulas for sums over arithmetic progressions of coefficients of products of the form

On coefficients of polynomials over finite fields

In this paper we study the relation between coefficients of a polynomial over finite field Fq and the moved elements by the mapping that induces the polynomial. The relation is established by a special system of linear equations. Using this relation we give the lower bound on the number of nonzero coefficients of polynomial that depends on the number m of moved elements. Moreover we show that there exist permutation polynomials of special form that achieve this bound when m|q−1. In the other direction, we show that if the number of moved elements is small then there is an recurrence relation among these coefficients. Using these recurrence relations, we improve the lower bound of nonzero coefficients when m?q−1 and . As a byproduct, we show that the moved elements must satisfy certain polynomial equations if the mapping induces a polynomial such that there are only two nonzero coefficients out of 2m consecutive coefficients. Finally we provide an algorithm to compute the coefficients of the polynomial induced by a given mapping with O(q3/2) operations.     

On infinite arithmetic progressions in sumsets

Let k be a positive integer.Denote by D_1/k the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference at most d,where(k+1)A is the set of all sums of k+1 elements(not necessarily distinct) of A.Chen and Li(2019) conjectured that D_1/k=k²+o(k²).The purpose of this paper is to confirm the above conjecture.We also prove that D_1/k is a ...     

On arithmetic progressions on Pellian equations

We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x ² ? dy ² = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.     

On products of consecutive arithmetic progressions. II

Let \({f(x, k, d) = x(x + d)\cdots(x + (k - 1)d)}\) be a polynomial with \({k \geq 2}\), \({d \geq 1}\). We consider the Diophantine equation \({\prod_{i = 1}^{r} f(x_i, k_i, d) = y^2}\), which is inspired by a question of Erd?s and Graham [4, p. 67]. Using the theory of Pellian equation, we give infinitely many (nontrivial) positive integer solutions of the above Diophantine equation for some cases.     

The average value of Fourier coefficients of cusp forms in arithmetic progressions

Recently Blomer showed that if α(n) denote the normalized Fourier coefficients of any holomorphic cusp form f with integral weight, then

     

Fourier coefficients of GL(N) automorphic forms in arithmetic progressions

We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all \({N \geq 2}\) , satisfy a central limit theorem in a suitable range, generalizing the case N = 2 treated by Fouvry et al. (Commentarii Math Helvetici, 2014). Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.     

On some Rado numbers for generalized arithmetic progressions

The 2-color Rado number for the equation x₁+x₂−2x₃=c, which for each constant we denote by S₁(c), is the least integer, if it exists, such that every 2-coloring, Δ : [1,S₁(c)]→{0,1}, of the natural numbers admits a monochromatic solution to x₁+x₂−2x₃=c, and otherwise S₁(c)=∞. We determine the 2-color Rado number for the equation x₁+x₂−2x₃=c, when additional inequality restraints on the variables are added. In particular, the case where we require x₂<x₃<x₁, is a generalization of the 3-term arithmetic progression; and the work done here improves previously established upper bounds to an exact value.