The mean square of real character sums

Restricted sums of four integral squares

We give a simple quaternionic proof of a recent result of Goldmakher and Pollack on restricted sums of four integral squares.

Binary forms as sums of two squares and Châtelet surfaces

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.

On supercongruences for truncated sums of squares of basic hypergeometric series

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.

On a Conjecture of Sun about sums of restricted squares

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.

On squares of sums of three cubes

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.

Sums of Two Squares Visualized

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.

Sums of squares in real quadratic fields and Hilbert modular groups

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.

Sums of Quaternion Squares and a Theorem of Watson

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.

Pairs of diagonal quadratic forms and linear correlations among sums of two 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.

Sums of two squares in short intervals

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.

关于某些带有模平方根的双线性和及其应用

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。

On the mean square of short exponential sums related to cusp forms

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.

关于带有模平方根的双线性和及其应用III

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。

On a conjecture for representations of integers as sums of squares and double shuffle relations

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.

Finiteness theorems for universal sums of squares of almost primes

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.

Numbers represented by restricted sums of four 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.

Sums of squares in Quaternion rings

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.

Lemke Oliver and Soundararajan bias for consecutive sums of two squares

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 …

关于带有模平方根的双线性和及其应用II

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。

分解为两平方和的分拆

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

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.

Sums of primes and squares of primes in short intervals

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.

可表示为两个平方数之和的整数有限序列

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。

Ternary Sums of Squares and Triangular Numbers

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.

三角数与平方数的泛性和

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。

On sums of squares of primes II

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.

Arithmetic functions commutable with sums of squares

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.

Mean square estimate for relatively short exponential sums involving Fourier coefficients of cusp forms

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.

On sums of three squares

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 $r_3(n)$.

Representing multiples of $m$ in real quadratic fields as sums of squares

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 $m \leq 5000$.

Ternary codes associated with symplectic groups and power moments of Kloosterman sums with square arguments

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).

Sums of integers and sums of their squares

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…

Perfect powers that are sums of squares in a three term arithmetic progression

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.

除固定整数外的整数平方和

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。

k项形式的平方和

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。

随机多项式值上 $\operatorname{GL}(m)$-自守形式系数和的平方根抵消

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。

Intervals between numbers that are sums of two squares

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 $γ\leq 2$.

Sums of four squares of primes

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 $11/32$.

关于素数加平方或立方之和的计算结果

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。

Sums of four squares with a certain restriction

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.

On integers which are representable as sums of large squares

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.

Salié和的移位双线性和与移位素数的模平方根分布

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。

Two-dimensional Weyl sums failing square-root cancellation along lines

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…

On the period of the continued fraction for values of the square root of power sums

The present paper proves that if for a power sum $α$ over $\ZZ$ the length of the period of the continued fraction for $\sqrt{α(n)}$ is constant for infinitely many even (resp. odd) $n$, then $\sqrt{α(n)}$ admits a functional continued fraction expansion for all even (resp. odd) $n$, except finitely many; in particular, for such $n$, the partial quotients can be expressed by power sums of the same kind.

The asymptotic formulae of sums of two smooth squares for divisor function

A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. Let $s_{1}$ and $s_{2}$ are $y$-smooth numbers. We consider sums of smooth squares of the binary Titchmarsh divisor problem and give asymptotic formulae for $\sum_{s_{1}^{2}+s_{2}^{2}\le x}τ(s_{1}^{2}+s_{2}^{2}+1)$ for $(\log x)^{K}\le y<x^{\frac{1}{2}}$, where $K$ is large enough.

两个平方数之和的连续序列的正密度

Positive density for consecutive runs of sums of two squares

本文研究了两个平方数之和的连续序列在等差数列中的分布，证明对于任意奇数无平方因子模数和简化同余类，存在正密度的此类连续序列，这是 Maynard 素数序列结果的类比。

Sums of squares in $S$-integers

In totally real number fields, we characterize the rings of $S$-integers (obtained by inverting a rational integer $m$) such that all their totally positive elements are represented as a sum of squares. We further obtain partial answers to the question: when are all the totally positive algebraic integers that are divisible by $m$ represented as a sum of squares?

Uchiyama's conjecture on sums of squares

Uchiyama showed that every interval $(n, n + c n^{1/4})$ contains an integer that is the sum of two squares, where $c= 2^{3/2}$. He also conjectured a minimal value of $c$ such that the above statement still holds. We investigate this claim.

Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

Considering the problem of finding all the integer solutions of the sum of $M$ consecutive integer squares starting at $a^{2}$ being equal to a squared integer $s^{2}$, it is shown that this problem has no solutions if $M\equiv3,5,6,7,8$ or $10 (mod\,12)$ and has integer solutions if $M\equiv0,9,24$ or $33 (mod\,72)$; or $M\equiv1,2$ or $16 (mod\,24)$; or $M\equiv11 (mod\,12)$. All the allowed values of $M$ are characterized using necessary conditions. If $M$ is a square itself, then $M\equiv1 (mod\,24)$ and $(M-1)/24$ are all pentagonal numbers, except the first two.

A sum of three nonunit squares of integers

We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann Hypothesis(GRH) holds. As applications, we find all integers, under the GRH only when $k=3$, which are sums of $k$ nonzero triangular numbers, sums of $k$ nonzero generalized pentagonal numbers, and sums of $k$ nonzero generalized octagonal numbers, respectively for any integer $k\ge 3$.

On Linnik's conjecture: sums of squares and microsquares

We show that almost all natural numbers n not divisible by 4, and not congruent to 7 modulo 8, are represented as the sum of three squares, one of which is the square of an integer no larger than (log n)^{1+e} (any e>0). This answers a conjecture of Linnik for almost all natural numbers, and sharpens a conclusion of Bourgain, Rudnick and Sarnak concerning nearest neighbour distances between normalised integral points on the sphere.

Binary linear forms as sums of two squares

We revisit recent work of Heath-Brown on the average order of the quantity r(L_1)r(L_2)r(L_3)r(L_4), for suitable binary linear forms L_1,..., L_4, for integers ranging over quite general regions. In addition to improving the error term in Heath-Brown's estimate we generalise his result quite extensively.

Kitaoka 猜想与平方和

Kitaoka's Conjecture and sums of squares

本文建立了全实域上三元经典万有二次型存在性与2的所有全正倍数为平方和之间的联系，并证明了Kitaoka猜想对奇判别域成立。

算术多边形与连续平方和

Arithmetic Polygons and Sums of Consecutive Squares

本文研究了算术多边形与连续平方和的关系，证明了对于奇数N≥3存在至少一个N边算术多边形，并展示了存在无限多个偶数边的算术多边形。