Guang-Liang Zhou, Yue-Feng She
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.
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We give a simple quaternionic proof of a recent result of Goldmakher and Pollack on restricted sums of four integral 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.
Character sums over elements of extensions of finite fields with restricted coordinates
本文研究了有限域扩张元素上具限制坐标的字符和问题，获得了乘性和加性特征和的非平凡界，并特别给出了Cantor集有限域类比上的非平凡估计。
We first study the mean value of certain restricted divisor sums involving the Chowla-Walum sums, improving in particular a recent estimate given by Iannucci. The aim of the second part of this work is the generalization of the previous study, by restricting the range of the divisors in the studied divisor sums, extending the Chowla-Walum conjecture, proving a small part of this extended conjecture and generalizing the asymptotic formulas previously obtained in the first part.
Restricted divisor functions and half Appell sums in higher-level
本文研究高阶半Appell和相关q-级数系数的值分布，证明其等于限制除数函数且可取任意非负整数，并证实了两个相关猜想。
Let $h\geq 2$ be a positive integer. For any subset $\mathcal{A}\subset \mathbb{Z}_n$, let $h^{\wedge}\mathcal{A}$ be the set of the elements of $\mathbb{Z}_n$ which are sums of $h$ distinct elements of $\mathcal{A}$. In this paper, we obtain some new results on $4^{\wedge}\mathcal{A}$ and $5^{\wedge}\mathcal{A}$. For example, we show that if $|\mathcal{A}|\geq 0.4045n$ and $n$ is odd, then $4^{\wedge}\mathcal{A}=\mathbb{Z}_{n}$; Under some conditions, if $n$ is even and $|\mathcal{A}|$ is close to $n/4$, then $4^{\wedge}\mathcal{A}=\mathbb{Z}_{n}$.
Weighted and Restricted Sum Formulas of Euler Sums
本文利用 Euler 和数的正则化双洗牌关系，统一证明了多重 zeta 值的求和公式及其变体，并在附录中以三个表格总结了包括新公式在内的多种结果。
Let $G$ be an additive abelian group. Let $A=\{a_{0}, a_{1},\ldots, a_{k-1}\}$ be a nonempty finite subset of $G$. For a positive integer $h$ satisfying $1\leq h\leq k$, we let \[h\hat{}_{\underline{+}}A:=\{Σ_{i=0}^{k-1}λ_{i} a_{i}: (λ_{0},λ_{1}, \ldots, λ_{k-1}) \in \{-1,0,1\}^{k},~Σ_{i=0}^{k-1}|λ_{i}|=h \},\] be the restricted signed sumset of $A$. The direct problem for the restricted signed sumset $h\hat{}_{\underline{+}}A$ is to find the minimum number of elements in $h\hat{}_{\underline{+}}A$ in terms of $|A|$. The inverse problem for $h\hat{}_{\underline{+}}A$ is to determine the struct…
In this paper we study restricted sum formulas involving alternating Euler sums which are defined by ζ(s_1,...,s_{d};ε_1,...,ε_d)=\sum_{n_1>...>n_d\ge 1}\frac{ε_1^{n_1}... ε_{d}^{n_d}}{n_1^{s_1}... n_d^{s_d}}, for all positive integers s_1,...,s_{d} and ε_1=\pm 1,..., ε_{d}=\pm 1 with (s_1,ε_1) unequal (1,1). We call w=s_1+...+s_{d} the weight and d the depth. When ε_j=-1 we say the jth component is alternating. We first consider Euler sums of the following special type: ξ(2s_1,...,2s_{d})=ζ(2s_1,...,2s_{d};(-1)^{s_1},...,(-1)^{s_{d}}). For d\le n, let Ξ(2n,d) be the sum of all ξ(2s_1,..., 2s_…
We give an upper bound for the minimum $s$ with the property that every sufficiently large integer can be represented as the sum of $s$ positive $k$-th powers of integers represented as the sum of three positive cubes for the cases $2\leq k\leq 4.$
Restricted sumsets in multiplicative subgroups
本文研究了有限域上乘法子群及整数完美幂中的受限和集，扩展了Shkredov的结果，并提出了一个受限和集的van Lint-MacWilliams猜想的类比。
We study an apparently new question about the behaviour of Weyl sums on a subset $\mathcal{X}\subseteq [0,1)^d$ with a natural measure $μ$ on $\mathcal{X}$. For certain measure spaces $(\mathcal{X}, μ)$ we obtain non-trivial bounds for the mean values of the Weyl sums, and for $μ$-almost all points of $\mathcal{X}$ the Weyl sums satisfy the square root cancellation law. Moreover we characterise the size of the exceptional sets in terms of Hausdorff dimension. Finally, we derive variants of the Vinogradov mean value theorem averaging over measure spaces $(\mathcal{X}, μ)$. We obtain general res…
Sums of four generalized polygonal numbers of almost prime length
研究四类受限质数因子数的广义多边形数之和，证明大数可表示为此类和，参数限制最多988个质因子。
A double-sum analogue of Whipple's summation formula and a restricted sum formula for multiple binomial sums
该研究证明了Whipple的{}_3F_2求和公式的双重和模拟式，并将其应用得到Igarashi参数化多重zeta级数特殊情形下多重二项和的受限求和公式
Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would follow if it were known for abc sums that are not highly divisible. We prove both theorems for general number fields.
A sequence $\bfx=(x_1,\ldots,x_m)$ of elements of $\Z_n$ is called an \textit{$A$-weighted Davenport Z-sequence} if there exists $\bfa:=(a_1,\ldots,a_m)\in (A\cup\{0\})^m\setminus\bfzero_m$ such that $\sum_i a_ix_i=0$. Here $\bfzero_m=(0,\ldots,0)\in\Z_n^m$. Similarly, the sequence $\bfx$ is called an \textit{$A$-weighted Erdős Z-sequence} if there exists $\bfa:=(a_1,\ldots,a_m)\in (A\cup\{0\})^m\setminus\{\bfzero_m\}$ with $|Supp(\bfa)|=n$, such that $\sum_i a_ix_i=0$, where $Supp(\bfa):=\{i: a_i\ne 0\}$. A $\Z_n$-sequence $\bfx$ is called $k$-restricted if no element of $\Z_n$ appears more t…
Consider the linear congruence equation $x_1+\ldots+x_k \equiv b\,(\text{mod } n)$ for $b,n\in\mathbb{Z}$. By $(a,b)_s$, we mean the largest $l^s\in\mathbb{N}$ which divides $a$ and $b$ simultaneously. For each $d_j|n$, define $\mathcal{C}_{j,s} = \{1\leq x\leq n^s | (x,n^s)_s = d^s_j\}$. Bibak et al. gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_i$. We generalize their result with generalized gcd restrictions on $x_i$ by proving that for the above linear congruence, the number of solutions is $$\frac{1}{n^s}\s…
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?
On restricted averages of Dedekind sums
本文研究固定$α\leq 1/2$时特定有理数集合上Dedekind和的平均值，将先前$α=1/2$的渐近结果扩展至所有固定有理数及几乎所有无理数$α$，并量化了by-excess Euclidean算法运行时间渐近式第二项的偏差。
Some direct and inverse problems for the Restricted Signed sumset in set of integers
该研究针对整数集合的受限h重符号和集的正逆问题，证明了h=4时相关猜想成立，还在集合约束下证明了任意h的正逆定理，并验证了猜想对等差数列成立。
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 estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we apply the "main conjecture" on the Vinogradov mean value theorem which has been established by Bourgain, Demeter and Guth (2016) as well as by Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets that are defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.
The difference of the sums of odd and even parts of restricted partitions
本文推导了在部分不重复的限制条件下，分拆的奇部和与偶部和之差的生成函数，并给出了模4的同余式。
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 investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be written as a ternary sumset. This proves a conjecture by Sárközy. We also clean up and sharpen existing results on sumset decompositions of the prime numbers.
We investigate the solubility of the congruence xy=1 (mod p), where p is a prime and x,y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.
On sums of unequal powers of primes and powers of 2
本文证明了充分大的偶数可表示为素数平方、立方、四次方及2的幂之和，且至少5.313%的奇数可表示为素数平方、立方与四次方之和，改进了R. Zhang的结果。
We prove an asymptotic formula for the number of S-integers in a number field K that can be represented by a sum of n S-integers of bounded norm.
Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a coset of a subgroup <g^s>, s|(q-1). We give the number explicitly for s=2, 3, 4 when q=p, and for s|(p^e+1) when r=2en. Finally, we give explicit formulae for the number when both trace and norm are fixed, p=2 and m =< 30.
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。
It is proved that all sufficiently large integers $n$ can be represented as $$n=x_1^2+x_2^3+\cdots+x_{13}^{14},$$ where $x_1,\ldots,x_{13}$ are positive integers. This improves upon the current record with $14$ variables in place of $13$.
Direct and Inverse Problems for Restricted Signed Sumsets -- II
本文研究受限h重带符号和集的直接与逆问题，在非负整数集合上完全证明了Bhanja等人提出的h≥4情形猜想，从而彻底解决了所有相关猜想。
We study decompositions of natural numbers into triangular summands. For instance, we prove that any natural number can be represented as a sum of four triangular numbers, two of them having even indices and the other two having odd indices. This settles a conjecture by Zhi-Wei Sun (arXiv:1503.03743v8).
We investigate the asymptotic formula for the number of representations of a large positive integer as a sum of $k$-th powers of integers represented as the sums of three positive cubes, counted with multiplicities. We also obtain a lower bound for the number of representations when the sums of three cubes are counted without multiplicities.
On restricted sums of four squares and Zhi-Wei Sun's $x+24y$ conjecture
本文利用三次二次型理论研究四平方定理的细化问题，并推进了齐桓-韦-孙的$x+24y$猜想。
Sums of generalized polygonal numbers of almost prime "length"
本文证明在$m$模30的限制下，密度为一的整数可表示为参数至多含6361个素因子的三个广义$m$边形数之和，并给出$f_m(n)$无平方部分足够大时的表示条件。
For a positive integer $h$ and a subset $A$ of a given finite abelian group, we let $hA$, $h \hat{\;} A$, and $h_{\pm}A$ denote the $h$-fold sumset, restricted sumset, and signed sumset of $A$, respectively. Here we review some of what is known and not yet known about the minimum sizes of these three types of sumsets, as well as their corresponding critical numbers. In particular, we discuss several new open direct and inverse problems.
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
On an inverse problem for restricted sumsets
本文研究了限制和集的反问题，在Liu和Sun于2002年给出的下界基础上，确定了当该下界达到时集合A的结构。
We give some restricted sum formulas for double zeta values whose arguments satisfy certain congruence conditions modulo 2 or 6, and also give an application to identities showed by Ramanujan for sums of products of Bernoulli numbers with a gap of 6.
Restricted sums of sets of cardinality $2p + 1$ in $\mathbb{Z}_p^2$
本文研究了Z_p^2中2p+1个元素集合的受限和问题，证明其基数至少为4p，解决了Erdős-Heilbronn问题的一个变种。
We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including $1$). For example, we show that each $n=1,2,3,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})$ with $|x+y-z|\in\{4^k:\ k\in\mathbb N\}$ (or $|2x-y|\in\{4^k:\ k\in\mathbb N\}$, or $x+y-z\in\{\pm 8^k:\ k\in\mathbb N\}\cup\{0\}\subseteq\{t^3:\ t\in\mathbb Z\}$), and that we can write any positive integer as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+2z$ (or $x+2y+2z$) a power of four. We also prove that any $n\in\mathbb N$ can be writte…
Rational approximation with digit-restricted denominators
本文证明了实数γ的“良好”逼近存在性，使用基b下由数字0和1构成分母的有理数，推导了基本估计并通过处理指数和增强该结果。
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.
The restricted partition function $p_{N}(n)$ counts the partitions of $n$ into at most $N$ parts. In the nineteenth century Sylvester showed that these partitions can be expressed as a sum of $k$-periodic quasi-polynomials ($1\leq k\leq N$) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called $q$-partial fractions) of the generating function of $p_{N}(n)$. In this paper we show that the coefficients of these $q$-partial fractions can be expressed as a linear combination of the Raman…
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$.
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 study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of Dirichlet $L$-functions and our proof relies on the asymptotic large sieve.
Diophantine approximation with sums of two squares
研究二元二次型在丢番图逼近中的应用，证明对于任意无理数α，存在无限多个由Q表示的正整数n，满足||αn||<n^(-1/2-ε)。同时给出当指数1/2-ε替换为任意固定γ<3/7时的结果量化版本。
In this article, we make use of some known method to investigate some properties of the numbers represented as sums of two equal odd powers, i.e., the equation $x^n+y^n=N$ for $n\ge3$. It was originated in developing algorithms to search new taxicab numbers (i.e., naturals that can be represented as a sum of positive cubes in many different ways) and to verify their minimality. We discuss properties of diophantine equations that can be used for our investigations. This techniques is applied to develop an algorithm allowing us to compute new taxicab numbers (i.e., numbers represented as sums of…
We obtain several asymptotic estimates for the sums of the restricted divisor function $$ τ_{M,N}(k) = #\{1 \le m \le M, \ 1\le n \le N: mn = k\} $$ over short arithmetic progressions, which improve some results of J. Truelsen. Such estimates are motivated by the links with the pair correlation problem for fractional parts of the quadratic function $αk^2$, $k=1,2,...$ with a real $α$.
The numbers of representations of totally positive integers as sums of three integer squares in $\mathbf{Q}(\sqrt{3})$ and in $\mathbf{Q}(\sqrt{17})$, are studied by using Shimura lifting map of Hilbert modular forms. We show the following results. In case of $\mathbf{Q}(\sqrt{3})$, a totally positive integer $a+b\sqrt{3}$ is represented as a sum of three integer squares if and only if $b$ is even. In case of $\mathbf{Q}(\sqrt{17})$, a totally positive integer is represented as a sum of three integer squares if and only if it is not in the form $π_{2}^{2e}π_{2}'^{2e'}μ$ with $μ\equiv7\pmod{π_{…
Some Generalizations of Totient Function with Elementary Symmetric Sums
本文基于初等对称多项式推广了欧拉函数，并推导出涉及第二个初等对称和的欧拉函数显式乘积形式，同时探讨了使用任意jth初等对称多项式时欧拉函数的行为，以及通过二次形式的最大公约数约束解决受限线性同余问题的方法。
Let $\mathcal{A}'$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p = m^2 + \ell^2$, with $\ell \in \mathcal{A}'$. The proof draws on ideas from the work of Friedlander-Iwaniec on primes of the form $p = x^2+y^4$, as well as ideas from the work of Maynard on primes with restricted digits.
A set of non-negative integers is an additive basis with range $n$, if its sumset covers all consecutive integers from 0 to $n$, but not $n+1$. If the range is exactly twice the largest element of the basis, the basis is restricted. Restricted bases have important special properties that facilitate efficient searching. With the help of these properties, we have previously listed the extremal restricted bases up to length $k = 41$. Here, with a more prudent use of the properties, we present an improved search algorithm and list all extremal restricted bases up to $k = 47$.
Sums of three cubes, II
本文改进迭代方法，给出 $4\le s\le 8$ 时三次光滑 Weyl 和 $s$ 次矩的估计，并由此改进不超过 $X$ 且可表为三个自然数立方和之整数个数的下界。
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.
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 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.
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