Song Guo, Hao Pan, Zhi-Wei Sun
For an integer $x$ let $t_x$ denote the triangular number $x(x+1)/2$. Following a recent work of Z. W. Sun, we show that every natural number can be written in any of the following forms with $x,y,z\in\Z$: $$x^2+3y^2+t_z, x^2+3t_y+t_z, x^2+6t_y+t_z, 3x^2+2t_y+t_z, 4x^2+2t_y+t_z.$$ This confirms a conjecture of Sun.
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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 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.
Indefinite theta functions arising from affine Lie superalgebras and sums of triangular numbers
本文通过推广 Roehrig 和 Zwegers 的不定 theta 函数理论，证明若干幂级数为模形式，并由此导出三角数生成函数幂的恒等式，进而揭示其与仿射李超代数分母恒等式的联系。
In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if T_m=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p^2=x^2+8(y^2+z^2) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2T_m (m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.
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).
In this paper, we prove that for $d=3,\dots,8$, every natural number can be written as $t_x+t_y+3t_z+dt_w$, where $x$, $y$, $z$, and $w$ are nonnegative integers and $t_k=k(k+1)/2$ $(k=0,1,2,\ldots)$ is a triangular number. Furthermore, we study mixed sums of triangular numbers and certain binary quadratic forms.
Triangular and tetrahedral number differences of sumset sizes in additive number theory
本文研究有限整数集合sumset大小的完整范围，发现在大小为4的集合流行sumset大小分布中存在与三角数和四面体数相关的意外模式。
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。
Sums of Reciprocals of Generalized Triangular Numbers
本文通过微积分方法计算了三角数倒数之和及其交替和，并推广到广义三角数推导闭式表达式。
We investigate here sums of triangular numbers $f(x):=\sum_i b_i T_{x_i}$ where $T_n$ is the $n$-th triangular number. We show that for a set of positive integers $S$ there is a finite subset $S_0$ such that $f$ represents $S$ if and only if $f$ represents $S_0$. However, computationally determining $S_0$ is ineffective for many choices of $S$. We give an explicit and efficient algorithm to determine the set $S_0$ under certain Generalized Riemann Hypotheses, and implement the algorithm to determine $S_0$ when $S$ is the set of all odd integers.
On a product of three theta functions and the number of representations of integers as mixed ternary sums involving squares, triangular, pentagonal and octagonal numbers
本文推导了三个theta函数乘积的线性组合一般公式，并由此得到两个theta函数乘积的公式，进而应用于整数表示为平方数、三角数、广义五角数和广义八边形数的混合三元和问题。
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.
For an arbitrary integer $x$, an integer of the form $T(x)=\frac{x^2+x}{2}$ is called a triangular number. For positive integers $α_1,α_2,\dots,α_k$, a sum $Δ_{α_1,α_2,\dots,α_k}(x_1,x_2,\dots,x_k)=α_1 T(x_1)+α_2 T(x_2)+\cdots+α_k T(x_k)$ of triangular numbers is said to be almost universal with one exception if the Diophantine equation $Δ_{α_1,α_2,\dots,α_k}(x_1,x_2,\dots,x_k)=n$ has an integer solution $(x_1,x_2,\dots,x_k)\in\mathbb{Z}^k$ for any nonnegative integer $n$ except a single one. In this article, we classify all almost universal sums of triangular numbers with one exception. Furth…
On periodic sign changes for weighted representations of integers as colored sums of triangular and generalized pentagonal numbers
研究整数表示为三角形数与三色广义五边形数之和的表示数的符号变化和消失，通过分析Dedekind-eta函数相关q级数展开的系数，并利用Hardy-Ramanujan-Rademacher圆法给出渐近公式。
Even universal sums of triangular numbers
本文分类了所有偶泛的三角数之和，并给出了任意三角数之和偶泛性的有效判定准则，推广了 Bosma 和 Kane 的三角数八定理。
On pairs of triangular numbers whose product is a perfect square and pairs of intervals of successive integers with equal sums of squares
本文研究了乘积为完全平方数的三角数对以及具有相等平方和的连续整数区间对，构造了多项式并提出了相关猜想。
We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitrary number as a sum of 4m^2/d triangles, whenever d divides 2m, and 4m(m+1)/d triangles, when d divides 2m or d divides 2m+2. This extends recent results of Getz and Mahlburg, Milne, and Zagier.
We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2^ap-r)/m+T_x with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with related analysis and numerical data. One of our conjectures states that each natural number not equal to 216 can be written in the form p+T_x with x an integer and p a prime or zero; another conjecture asser…
In this paper we study mixed sums of primes and linear recurrences. We show that if m=2(mod 4) and m+1 is a prime then $(m^{2^n-1}-1)/(m-1)\not=m^n+p^a$ for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer, u_0=0, u_1=1 and u_{i+1}=au_i+u_{i-1} for i=1,2,3,..., then all the sums u_m+au_n (m,n=1,2,3,...) are distinct. One of our conjectures states that any integer n>4 can be written as the sum of an odd prime, an odd Fibonacci number and a positive Fibonacci number.
In this paper we consider a variety of mixed character sums. In particular we extend a bound of Heath-Brown and Pierce to the case of squarefree modulus, improve on a result of Chang for mixed sums in finite fields, we show in certain circumstances we may improve on some results of Pierce for multidimensional mixed sums and we extend a bound for character sums with products of linear forms to the setting of mixed sums.
In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$, equivalently the form $2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $T_z$ denotes the triangular number $z(z+1)/2$. Given positive integers $a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ax^2+by^2+cT_z$ represents sufficiently large integers and establish similar results for the forms $ax^2…
On Nathanson's Triangular Number Phenomenon
本文利用格的逐次L1-极小值刻画整数有限集合的和集大小序列，解释其中三角数出现规律，并给出前两个极小值区间内的精确公式，从而阐明Nathanson实验中的现象。
Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A new representation for the generating function for sums of three triangular numbers is derived, which is slightly different from that of Andrews, also implies the famous result of Gauss where every integer is the sum of three triangular numbers.
This paper proves nontrivial bounds for short mixed character sums by introducing estimates for Vinogradov's mean value theorem into a version of the Burgess method.
Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\ldots,a_k,n\in\Bbb N$ let $N(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by $a_1x_1^2+a_2x_2^2+\cdots+a_kx_k^2$, and let $t(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by $a_1\frac{x_1(x_1-1)}2+a_2\frac{x_2(x_2-1)}2+\cdots+a_k\frac{x_k(x_k-1)}2$ $(x_1,\ldots,x_k\in\Bbb Z$). In this paper, by using Ramanujan's theta functions $φ(q)$ and $ψ(q)$ we reveal many relations between $t(a_1,a_2,\ldots,a_k;n)$ and $N(a_1,a_2,\ldots,a_k;8n+a_1+\cdots+a_k)$ f…
We establish bounds of triple exponential sums with mixed exponential and linear function. The method we use is by Shparlinski together with a bound of additive energy from Roche-Newton, Rudnev and Shkredov.
For non-negative integers $a,b,$ and $n$, let $N(a, b; n)$ be the number of representations of $n$ as a sum of squares with coefficients $1$ or $3$ ($a$ of ones and $b$ of threes). Let $N^*(a,b; n)$ be the number of representations of $n$ as a sum of odd squares with coefficients $1$ or $3$ ($a$ of ones and $b$ of threes). We have that $N^*(a,b;8n+a+3b)$ is the number of representations of $n$ as a sum of triangular numbers with coefficients $1$ or $3$ ($a$ of ones and $b$ of threes). It is known that for $a$ and $b$ satisfying $1\leq a+3b \leq 7$, we have $$ N^*(a,b;8n+a+3b)= \frac{2}{2+{a\ch…
This work proves a Burgess bound for short mixed character sums in $n$ dimensions. The non-principal multiplicative character of prime conductor $q$ may be evaluated at any "admissible" form, and the additive character may be evaluated at any real-valued polynomial. The resulting upper bound for the mixed character sum is nontrivial when the length of the sum is at least $q^β$ with $β> 1/2 - 1/(2(n+1))$ in each coordinate. This work capitalizes on the recent stratification of multiplicative character sums due to Xu, and the resolution of the Vinogradov Mean Value Theorem in arbitrary dimension…
Let $p$ be a prime number. Every two-variable polynomial $f(x_1, x_2)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of surfaces whose Galois group is isomorphic to $\mathbb Z_p$. Our goal of this paper is to study the Newton polygon of the $L$-functions associated to a finite character of $\mathbb{Z}_p$ and a generic polynomial whose convex hull is a fixed triangle $Δ$. We denote this polygon by $\textrm{GNP}(Δ)$. We prove a lower bound of $\textrm{GNP}(Δ)$, which we call the improved Hodge polygon $\textrm{IHP}(Δ)$, and we conjecture that $\textrm{GNP}(Δ)$ …
In this paper, we consider sums of class numbers of the type $\sum_{m\equiv a\pmod{p}} H(4n-m^2)$, where $p$ is an odd prime, $n\in \mathbb{N},$ and $a\in \mathbb{Z}$. By showing that these are coefficients of mixed mock modular forms, we obtain explicit formulas. Using these formulas for $p=5$ and $7$, we then prove a conjecture of Brown et al. in the case that $n=\ell$ is prime.
For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary quadratic forms to determine completely when the mixed sum $ax^2+by^2+cP_{p^k+2}(z)$ represents all but finitely many positive integers.
This paper proves Burgess bounds for short mixed character sums in multi-dimensional settings. The mixed character sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial, and a product of multiplicative Dirichlet characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation-dilation invariant systems in order to prove character sum bounds in any dimension that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding the given polynomial into an adv…
Consecutive Cycle Sums
本文研究将数字1至n排列在n-循环上，哪些数可表示为连续数字之和，并证明当n≥8时，n-循环不完备。
Prime Solutions to a Binary Additive Equation and Mixed Moments of Character Sums
本文通过泊松求和与Kloosterman分数和均值估计，得到二元加法方程素数解的渐近公式，并应用于特征和混合四阶矩，获得非平凡界。
Representing integers as sums of mixed powers of primes
该研究证明了每个足够大的奇整数可以表示为素数的混合次幂之和，其中包含平方、立方和更高次幂的组合。
Mixed incomplete character sums of rational functions with smooth moduli
本文研究了有理函数的混合不完整特征和，改进了平滑参数的界，并探讨了其推广与应用。
We express a family of Hecke--Appell-type sums of Hikami and Lovejoy in terms of mixed mock modular forms; in particular, we express the sums in terms of Appell functions and theta functions. Hikami and Lovejoy's family of Hecke--Appell-type sums was obtained by considering certain duals of generalized quantum modular forms.
The Möbius function of the poset of triangular numbers under divisibility
本文研究了整除关系下三角形数偏序集上的默比乌斯函数，通过Hasse图和矩阵可视化分析其部分和与经典默比乌斯函数的异同，并提出了关于其渐近行为和上下界的新猜想。
An integer of the form $P_m(x)= \frac{(m-2)x^2-(m-4)x}{2}$ for an integer $x$, is called a generalized $m$-gonal number. For positive integers $α_1,\dots,α_u$ and $β_1,\dots,β_v$, a mixed sum $Φ=α_1P_4(x_1)+\cdots+α_uP_4(x_u)+β_1P_8(y_1)+\cdots+β_vP_8(y_v)$ of generalized $4$- and $8$-gonal numbers is called universal if $Φ=N$ has an integer solution for any nonnegative integer $N$. In this article, we prove that there are exactly 1271 proper universal mixed sums of generalized $4$- and $8$-gonal numbers. Furthermore, the "$61$-theorem" is proved, which states that an arbitrary mixed sum of ge…
In \cite{ono}, K. Ono, S. Robins and P.T. Wahl considered the problem of determining formulas for the number of representations of a natural number $n$ by a sum of $k$ triangular numbers and derived many applications, including the one connecting these numbers with the number of representations of $n$ as a sum of $k$ odd square integers. They also obtained an application to the number of lattice points in the $k$-dimensional sphere. In this paper, we consider triangular numbers with positive integer coefficients. First we show that if the sum of these coefficients is a multiple of $8$, then th…
Large values of shifted mixed character sums
研究大偏移混合特征和的界，扩展了Montgomery和Iggidr的结果，允许一般非主特征和考虑不完整混合特征和。
Mixed moments of arithmetic functions and Kloosterman sums
本文推广Nair--Tenenbaum估计至Kloosterman和的混合矩，证明其渐近公式，并给出Hecke特征值混合矩上界、大Kloosterman值频率及谱计数方差下界。
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 $k\ge 3$.
We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for fixed positive integers $b_1, b_2,..., b_k$, represents every nonnegative integer if and only if it represents 1, 2, 4, 5, and 8. Moreover, if `cross-terms' are allowed in $f$, we show that no finite set of positive integers can play an analogous role, in turn showing that there is no overarching finiteness theorem which generalizes the statement from positive…
Let $0< n,α,β\in\mathbb{N}$ be such that $\gcd{(α,β)}=1$. We carry out the evaluation of the convolution sums $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {α\,k+β\,l=n} } }{\sum}σ(k)σ_{3}(l)$ and $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {α\,k+β\,l=n} } }{\sum}σ_{3}(k)σ(l)$ for all levels $αβ\in\mathbb{N}$, by using in particular modular forms. We next apply convolution sums belonging to this class of levels to determine formulae for the number of representations of a positive integer $n$ by the quadratic forms in twelve variables $\underset{i=1}{\overset{12}{\sum}}x_{i}^{2}$ when…
Average sizes of mixed character sums
本文研究了满足弱迪奥菲安条件的无理实数θ下混合特征和的平均大小，证明其为\sqrt{x}，并利用二次迪奥菲安方程进行分析。
An estimate for incomplete mixed character sums and applications
本文通过函数域方法，对不完全混合特征和给出了非平凡界，并证明了在特定条件下有限域扩展具有弱直线性质。
Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions
本文利用Ramanujan theta函数恒等式理论，系统确定了由广义3-、4-、5-和8-角数构成的四元混合和何时能表示所有非负整数，即其通用性。
A chain is an ordering of the integers 1 to n such that adjacent pairs have sums of a particular form, such as squares, cubes, triangular numbers, pentagonal numbers, or Fibonacci numbers. For example 4 1 2 3 5 form a Fibonacci chain while 1 2 8 7 3 12 9 6 4 11 10 5 form a triangular chain. Since 1 + 5 is also triangular, this latter forms a triangular necklace. A search for such chains and necklaces can be facilitated by the use of paths of billiard balls on a rectangular or other polygonal billiard table.
Critical Pairs for Mixed Restricted Sumsets in Prime Fields
本文对素数域中混合受限和集达到最小可能大小的集合对进行了完全分类，主要贡献在于内部区间情形，证明了临界性迫使集合为公共端点等差数列，并处理了特殊仿射轨道及删除模式。
We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution which comes from the asymptotic mixed distribution of Salié sums.
In this article, we collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. We start with a sketch of Milne's proof of two of these conjectures. We also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of $4s^2$, respectively $4s(s+1)$, triangular numbers. This approach is inspired by Zagier's proof of the Kac--Wakimoto formulas via modular forms. We end the survey with recent conjectures of the first author and Chua.
On Sums of Practical Numbers and Polygonal Numbers
本文证明了所有正整数均可表示为实用数与三角形数之和，解决了 Sun 的猜想，并证明充分大的自然数可表示为实用数与两个 $s$ 角数之和。
Two-term dilogarithms, mixed-base Nahm sums, and Fricke symmetry
本文研究两项Rogers二对数关系对秩二Nahm系统的算术约束，确定特定族中的允许步长，证明指标13系统的两项恒等式并精确再现Fricke振幅，同时指出相关乘积求值仍为猜想。
In this article, we consider the problem of determining formulas for the number of representations of a natural number $n$ by a sum of figurate numbers with certain positive integer coefficients. To achieve this, we prove that the associated generating function gives rise to a modular form of integral weight under certain condition on the coefficients when even number of higher figurate numbers are considered. In particular, we obtain modular property of the generating function corresponding to a sum of even number of triangular numbers with coefficients. We also obtain modularity property of …
Distribution of mixed character sums and extremal problems for Littlewood polynomials
本文研究混合特征和的分布特性，并应用结果构造具有大Mahler测度和L1范数的Littlewood多项式，同时证明Turyn多项式L_{2k}范数在α=1/4处渐近最小，解决了Günther和Schmidt的猜想。
Finite $q$-multiple harmonic sums at roots of unity: Symmetrized identities for $1$-$2$ index multisets
本文研究了在 $2-\cdots-2,1-\cdots-1$ 指数下 $q$-多重调和和的显式表达式，探讨了指数组比值变化时的更一般结果。
Mixed character sums modulo prime powers
本文研究了模素数幂的混合特征和的显式估计，通过分析多项式和有理函数的度数及零点，推导出不同条件下特征和的上界。
In this paper we introduce confluence relations for motivic Euler sums (also called alternating multiple zeta values) and show that all linear relations among motivic Euler sums are exhausted by the confluence relations. This determines all automorphisms of the de Rham fundamental groupoid of $\mathbb{P}^{1}\setminus\{0,\pm1,\infty\}$ coming from the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$. Moreover, we also discuss other applications of the confluence relations such as an explicit $\mathbb{Q}$-linear expansion of a given motivic Euler sum by their basis…
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