Agustin Moreno
We prove that a positive integer not of the form, 4^{k}(8m+7) can be expressible as a sum of three or fewer squares by using some results of Kane and Sun on mixed sums of squares and triangular numbers.
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In this paper, we obtain formulas for the number of representations of positive integers as sums of arbitrarily many squares (and other polygonal numbers) with a certain natural weighting. The resulting weighted sums give Fourier coefficients of weight two quasimodular forms.
A finite set $A$ of integers is square-sum-free if there is no subset of $A$ sums up to a square. In 1986, Erd\H os posed the problem of determining the largest cardinality of a square-sum-free subset of $\{1, ..., n \}$. Answering this question, we show that this maximum cardinality is of order $n^{1/3+o(1)}$.
By means of $q$-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus $x^2+y^2$ for some integers $x$ and $y$ with $x\not\equiv y (mod 2)$ or $x=y>0$. The paper also contains some other results and open conjectures on mixed sums of squares and triangular numbers.
We classify the connection between $t$-cores and self-conjugate $t$-cores to sums of squares. To do so, we provide explicit maps between $t$-core partitions and self-conjugate $t$-core partitions of a positive integer $n$ to representations of certain numbers as sums of squares. For example, the self-conjugate $4$-core partition $λ=(4,1,1,1)$ corresponds uniquely to the solution $61=6^2+5^2$. As a corollary, we completely classify the relationship between $t$-cores and Hurwitz class numbers. Using these tools, we see how certain sets of representations as sums of squares naturally decompose in…
It is demonstrated how an explicit expression of the (partial) sum of Tetranacci numbers can be found and proved using generating functions and the Hadamard product. We also provide a Binet-type formula for generalized Fibonacci numbers, by explicitly factoring the denominator of their generating functions.
We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of the coefficients of some infinite products.
Sums of Kloosterman sums over square-free and smooth integers
本文受近期关于 Kloosterman 和的双线性型及扭曲和的研究启发，研究了由无平方因子整数和光滑整数参数化的 Kloosterman 和之和的若干新问题。
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
We prove that the sequence of the sums of two squares do not have metric Poissonian pair correlation.
We give a bijective parameter representation for a sum of squares of numbers being equal to another sum of squares of numbers.
We use Bourgain's recent bound for short exponential sums to prove certain independence results related to the distribution of squarefree numbers in arithmetic progressions.
We determine all perfect powers that can be written as the sum of at most 10 consecutive squares.
We give a simple quaternionic proof of a recent result of Goldmakher and Pollack on restricted sums of four integral squares.
We investigate the frequency of positive squareful numbers x,y,z for which x+y=z.
The representation of integral binary forms as sums of two squares is discussed and applied to establish the Manin conjecture for certain Châtelet surfaces defined over the rationals.
Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of certain basic hypergeometric series. In this note we extend some of these supercongruences to the truncated sums of the squares of the corresponding series.
In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms $2^a3^b$ and $2^c5^d$. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem.
We show that almost every positive integer can be expressed as a sum of four squares of integers represented as the sums of three positive cubes.
We provide a geometric interpretation of Brillhart's celebrated algorithm for expressing a prime $p\equiv 1\pmod 4$ as the sum of two squares.
We use the spectral theory of Hilbert-Maass forms for real quadratic fields to obtain the asymptotics of some sums involving the number of representations as a sum of two squares in the ring of integers.
We use a representability theorem of G. L. Watson to examine sums of squares in Quaternion rings with integer coefficients. This allows us to determine a large family of such rings where every element expressible as the sum of squares can be written as the sum of 3 squares.
For suitable pairs of diagonal quadratic forms in 8 variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares.
We study the representation theory of a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of "magical" combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.
We show that there are short intervals $[x,x+y]$ containing $\gg y^{1/10}$ numbers expressible as the sum of two squares, which is many more than the average when $y=o( (\log{x})^{5/9})$. We obtain similar results for sums of two squares in short arithmetic progressions.
On certain bilinear sums with modular square roots and applications
扩展了Dunn等人关于模平方根加法能量的界，并将结果应用于带模平方根的双线性指数和，在平方模大筛法上取得部分进展。
The purpose of the article is to estimate the mean square of a squareroot length exponential sum of Fourier coefficients of a holomorphic cusp form.
On bilinear sums with modular square roots and applications III
本文改进了之前方法，针对素数平方模，通过限制二次高斯和到简化剩余类，实现显著消去，提升双线性和的分析。
In this paper, we prove a conjecture of Chan and Chua for the number of representations of integers as sums of 8s integral squares. The proof uses a theorem of Imamoglu and Kohnen, and the double shuffle relations satisfied by the double Eisenstein series of level 2.
In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.
We give a new proof of Milne's formulas for the number of representations of an integer as a sum of 4m^2 and 4m(m+1) squares. The proof is based on explicit evaluation of pfaffians with elliptic function entries, and relates Milne's formulas to Schur Q-polynomials and to correlation functions for continuous dual Hahn polynomials. We also state a new formula for 2m^2 squares.
In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$ where $x,y,z,t$ are integers and $x+y+2z+2t$ is a square or a cube.
Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with integer coefficients. We determine the minimum necessary number of squares for infinitely many Quaternion rings, and give global upper and lower bounds.
In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy--Littlewood conjectures containing a large secondary term, which fits the data very well. In this paper, we study consecutive pairs of sums of squares in arithmetic progressions, and develop a similar heuristic model based on the Hardy--Littlewood conjecture for sums of squares, which also explain the biases in the experimental data. In the process, we prove several …
On bilinear sums with modular square roots and applications II
本文进一步研究了带有模平方根的双线性和和加性能量，改进了变量范围较大时的结果，并在平方模数的大筛理论中取得进展。
Partitions into Sums of Two Squares
本研究通过圆法变体结合鞍点法，解决了两平方和整数分拆数的渐近公式问题，突破了以往受限分拆研究局限，首次处理了带分数奇点的狄利克雷级数
Sums of squares of $|ζ(1/2+it)|$ over short intervals are investigated. Known upper bounds for the fourth and twelfth moment of $|ζ(1/2+it)|$ are derived. A discussion concerning other possibilities for the estimation of higher power moments of $|ζ(1/2+it)|$ is given.
Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. We prove that when $H \ge X^{0.33}$, almost all integers $n \in \mathbf H_2$, $X < n \le X + H$ can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.
Finite sequences of integers expressible as sums of two squares
本文构造了无穷多组满足特定间隔条件的整数序列，使其各项均可表示为两个非零整数平方和，并给出连续三项、四项、五项及公差为4的五项算术级数等参数化结果。
For any integer $x$, let $T_x$ denote the triangular number $\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all the triples of positive integers $(α, β, γ)$ for which the ternary sums $αx^2 +βT_y + γT_z$ represent all but finitely many positive integers. This resolves a conjecture of Kane and Sun \cite[Conjecture 1.19(i)]{KS08} and complete the characterization of all almost universal ternary mixed sums of squares and triangular numbers.
Universal Sums of Triangular Numbers and Squares
本文研究三角数与平方数的泛性和问题，证明其泛性等价于能表示一组给定的正整数。
In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exceptional sets.
We describe the structure of a set of integers $A$ of positive density $δ$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $δ_0=1/4-\frac{2}{π^2}=0.0473...$ and $δ_1=1/18$.
In this note, we characterize all functions $f : \mathbb{N} \rightarrow \mathbb{C}$ such that $f(x_1^2+ \cdots + x_k^2)=f(x_1)^2+ \cdots + f(x_k)^2$, where $k \geq 3$ and $x_1, \cdots, x_k$ are positive integers.
We estimate the mean square of a short exponential sum involving Fourier coefficients of a cusp form with a linear twist, a smooth weight function, and a relatively short averaging interval.
Let $r_3(n)$ be the number of representations of a positive integer $n$ as a sum of three squares of integers. We give two distinct proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of $r_3(n)$.
We study real quadratic fields $\mathbb{Q}(\sqrt{D})$ such that, for a given rational integer $m$, all $m$-multiples of totally positive integers are sums of squares. We prove quite sharp necessary and sufficient conditions for this to happen. Further, we give a fast algorithm that solves this question for specific $m$, $D$ and we give complete results for $m \leq 5000$.
In this paper, we construct two ternary linear codes associated with the symplectic groups Sp(2,q) and Sp(4,q). Here q is a power of three. Then we obtain recursive formulas for the power moments of Kloosterman sums with square arguments and for the even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of "Gauss sums" for the symplectic groups Sp(2n,q).
Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares of $m$ integers? Denoting this set of all possible $T$ by $\mathscr{S}_m(n)$, Goldmakher and Pollack have given a simple characterization of $\mathscr{S}_4(n)$ using elementary arguments. Their result can be reinterpreted in terms of Mordell's theory of representations of binary integral quadratic forms as sums of squares of integral linear f…
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 \le r \le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
Sums of squares of integers except for a fixed one
本文研究除固定整数外的整数平方和问题，对任意非负整数$n$找到了能表示所有相关正整数的最小平方数数量，推广了Dubouis在$n=0$时的结果。
In this paper, we evaluate a smoothed sum of the form $\displaystyle \sideset{}{^*}\sum_{d \leq X}\left(\sum_{n \leq Y} \left(\frac{8d}{n} \right ) \right)^2$, where $\left ( \frac{8d}{\cdot} \right )$ is the Kronecker symbol and $\sideset{}{^*}\sum$ denotes a sum over positive odd square-free integers.
Sums of squares of k-term forms
本文研究至多k项的n元d次形式的平方和锥，证明其随k严格嵌套，并构造了具有特定项数的不定不可约形式。
Square-root cancellation for sums of coefficients of $\operatorname{GL}(m)$-automorphic forms over values of random polynomials
本文证明了对任意 $m\geq 2$ 和 $d\geq 1$，$\mathrm{GL}(m)$-自守形式的傅里叶系数在随机 $d$ 次多项式值上的平均和具有平方根抵消。
In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider certain sum of Bessel functions and prove the upper bound for its weighted mean value. This bound provides estimates for the $γ$-th moments of gaps for all $γ\leq 2$.
Let $E(N)$ denote the number of positive integers $n \le N$, with $n \equiv 4 \pmod{24}$, which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$.
Computational results on sums of a prime with squares or cubes
本文通过计算验证了素数加两个平方数或立方数的表示猜想，证明了特定范围内所有整数均可表示，并利用椭圆曲线整点方法实现。
In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$ with $x+3y$ a square. Meanwhile, he also conjectured that for each positive integer $n$ there exist integers $x,y,z,w$ such that $n=x^2+y^2+z^2+w^2$ and $x+3y\in\{4^k:k\in\mathbb{N}\}$. In this paper, we confirm these conjectures via some arithmetic theory of ternary quadratic forms.
We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set $\{n^2,(n+1)^2,\ldots \}$ is asymptotically $O(n^2)$, verifying thus a conjecture of Dutch and Rickett. Furthermore we ask a question on the representation of integers as sum of four large squares.
Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes
研究Salié和的移位双线性和的上界，并分析移位素数的模平方根分布，揭示$b\not \equiv 0 \bmod q$时的新困难。
We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a…
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