jorgen cherly
We obtain using exponential quadratic sums, explicit expressions for the number of triple persymmetric matrices over F_2 of given rank. (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j = r+s)
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In this paper we expose our main results about rank problems concerning persymmetric matrices over F_2 associated to some exponential sums.
We obtain, using exponential quadratic sums, explicit expressions for the number of double persymmetric matrices with entries in F_2 of given rank. (A matix [a(i,j)) is persymmetric if a(i,j) = a(r,s) for i+j = r+s)
Over the finite field with two elements, we present a method for obtaining explicit expressions for the number of rank i matrices of the form A above B, where A is persymmetric (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j = r+s).
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.
Triple sums of Kloosterman sums and the discrepancy of modular inverses
研究大区间内模正整数$c$的模逆分布，给出其盒、球及各向同性偏差的上下界以展示与随机点集的偏离，分析基于克洛斯特曼和三重和的新界。
Inspired by the recent work of El Bachraoui, we present some new $q$-supercongruences on triple and quadruple sums of basic hypergeometric series. In particular, we give a $q$-supercongruence modulo the fifth power of a cyclotomic polynomial, which is a $q$-analogue of the quadruple sum of Van Hamme's supercongruence (G.2).
We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some applications to primes in short intervals.
In this work we provide a meromorphic continuation in three complex variables of two types of triple shifted convolution sums of Fourier coefficients of holomorphic cusp forms. The foundations of this construction are based in the continuation of the spectral expansion of a special truncated Poincaré series recently developed by Jeffrey Hoffstein. As a result we are able to produce previously unstudied and nontrivial asymptotics of truncated shifted sums which we expect to correspond to off-diagonal terms in the third moment of automorphic L-functions.
A heuristic guide to evaluating triple-sums
本文提出借助联系 Appell–Lerch 函数与发散偏 theta 函数的启发式方法，将 Hecke 型二重和展开，并通过示例指导类似三重和的计算。
Let $τ_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}τ_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log x)^2+ c_2x^{\frac{3}{2}}\log x +c_3x^{\frac{3}{2}} +O_{\varepsilon}(x^{\frac{11}{8}+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$.
Triple convolution sums of the generalised divisor functions and related sums over primes
本文研究广义除数函数的三重卷积和及其在素数上的应用，通过多变量Dirichlet级数理论和概率论方法推导出渐进行为及下界。
In this paper, we study the alternating Euler $T$-sums and $§$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $§$-sums, from which, the parity theorems of Hoffman's double and triple $t$-values and Kaneko-Tsumura's double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $§$-sums are expressible in…
Triple Correlation Sums of Coefficients of Cusp Forms
本文研究全纯尖点形式 Fourier 系数的三重平移关联和，给出无条件渐近估计及 Riemann 假设下的加强，并据此证明无穷多个三项等差数列上系数乘积非零。
We prove that a certain family of sums of products of three binomials has alternating behavior modulo a prime $p$. To accomplish this we rewrite these sums as signed sums of products of three binomials, the better to handle $p$, and we give closed-form expressions for two related sums of signed products of three binomials.
We give a new bound on colinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.
In this paper, we investigate a number of $q$-supercongruences on double and triple sums. By means of a lemma devised by El Bachraoui and its generalization, we transform some $q$-supercongruences on double and triple sums into the $q$-supercongruences of the square and cube of truncated basic hypergeometric series.Our main tools still involve the `creative microscoping' method introduced by Guo and Zudilin, a lemma designed by Guo and Li and the Chinese remainder theorem.
We give a new bound on the number of collinear triples for two arbitrary subsets of a finite field. This improves on existing results which rely on the Cauchy inequality. We then us this to provide a new bound on trilinear and quadrilinear exponential sums.
The explicit formulas for the sums of positive powers of the integers $s_i$ unrepresentable by the triple of integers $d_1,d_2,d_3\in {\mathbb N}, \gcd(d_1,d_2,d_3)=1$, are derived.
In this paper, we study the average size of the product of a number and its multiplicative inverse modulo a prime p. This turns out to be related to moments of L-functions and leads to a curious asymptotic formula for a certain triple exponential sum.
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.
Rogers-Ramanujan type identities involving double, triple and quadruple sums
本文使用Maple搜索发现了多个涉及二重、三重和四重和的Rogers-Ramanujan型恒等式，并通过直接求和或常数项法将其转化为已知恒等式进行证明，同时得出新的单重和恒等式。
Proof of Kaneko--Tsumura Conjecture on Triple T-Values
本文利用 Euler 和生成函数，将多重 zeta 值的加权和关系推广至 Euler 和等变体，重新证明已知公式，证实 Kaneko–Tsumura 三重 T 值猜想并发现新恒等式。
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。
A fourth derivative test for exponential sums
本文通过将指数和问题归结为三重指数和的均方值定理，把经典 van der Corput 指数 1/14 改进为 1/13，给出了依赖四阶导数大小的新上界。
The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\ (n=0,1,2,\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\ldots$ can be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat.
We show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan Delta-function $Δ(τ)=η^{24}(τ)$, $η^{12}(τ)$ and $η^{8}(τ)$ in terms of Cohen's numbers $H(3,N)$ and $H(5,N)$. We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the J…
We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical object of study of additive combinatorics.
We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$) involves a linear combination of $n$ multiple harmonic sums, and holds $\mod{p^{2n+3}}$. The coefficients in these congruences are integers depending on $n$ and $k$, but independent of $p$. More generally, we construct a family of congruences for ${kp-1\choose p-1} \mod{p^{2n+3}}$, whose members contain a variable number of terms, and show that in this family …
Mizuno's rank three Nahm sums I: identities of index $(1,1,2)$
本文确认了Mizuno提出的19个秩三Nahm和的模性，通过建立指标$(1,1,2)$的Rogers--Ramanujan型恒等式，并推广了多重和恒等式。
We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new bounds on trilinear exponential sums and on the total number of residues equaling the product of two differences of elements of $\mathcal{A}$. We also prove a refined estimate on the number of collinear triples in a Cartesian product of multiplicative subgroups and derive stronger bounds for trilinear sums with all variables in multiplicative subgroups.
For $m=3,4,\ldots$, the polygonal numbers of order $m$ are given by $p_m(n)=(m-2)\binom n2+n\ (n=0,1,2,\ldots)$. For positive integers $a,b,c$ and $i,j,k\ge3$ with $\max\{i,j,k\}\ge5$, we call the triple $(ap_i,bp_j,cp_k)$ universal if for any $n=0,1,2,\ldots$ there are nonnegative integers $x,y,z$ such that $n=ap_i(x)+bp_j(y)+cp_k(z)$. We show that there are only 95 candidates for universal triples (two of which are $(p_4,p_5,p_6)$ and $(p_3,p_4,p_{27})$), and conjecture that they are indeed universal triples. For many triples $(ap_i,bp_j,cp_k)$ (including $(p_3,4p_4,p_5),(p_4,p_5,p_6)$ and $…
We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring $F_{q}[T]$ of polynomials over a finite field $F_{q}$ of $q$ elements, in the limit $q\rightarrow\infty$. We use a recent equidistribution result due to N. Katz to express these variances in terms of triple matrix integrals over the unitary group, and evaluate them.
We calculate the triple correlations for the truncated divisor sum $λ_{R}(n)$. The $λ_{R}(n)$'s behave over certain averages just as the prime counting von Mangoldt function $Λ(n)$ does or is conjectured to do. We also calculate the mixed (with a factor of $Λ(n)$) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation $Λ_{R}(n)$. However, when $λ_{R}(n)$ is used the error in the singular series app…
On a Transformation of Triple $q$-Series and Rogers-Hecke Type Series
本研究利用$q$-指数微分算子推广Sears $_4ϕ_3$变换公式，结合解析函数原点$q$-级数展开得到三重$q$-级数变换公式，并将其应用于Rogers-Hecke型级数，获得100多个相关恒等式。
On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-$p$ digits
该研究在$p$进数理论框架下给出Carmichael数的新刻画，揭示其与Bernoulli多项式分母、基$p$数字和的关联，定义了本原Carmichael数，证明Carmichael数可表为显式多边形数，并给出相关集合的模子集覆盖。
Proofs of Some Kanade--Russell Mod 12 Conjectures
本文用线性递推、q-差分方程、q-级数求和及围道积分与留数演算等方法，证明了Kanade--Russell模12猜想中剩余的六个恒等式，包括两个三重和与四个四重和恒等式。
Exact Frequencies of Consecutive Quadratic Residue and Nonresidue Patterns Modulo a Prime
研究模奇素数\(p\)时长度为二和三的连续二次剩余与非剩余模式的精确频率，利用有限域特征和基本性质推导计数公式，对的频率由\(p \pmod 4\)分类，三元组频率依赖\(p \pmod 8\)及雅可比和\(T_p\)。
Proofs of Mizuno's Conjectures on Rank Three Nahm Sums of Index $(1,2,2)$
该研究利用Bailey对理论和$q$-级数技巧，通过建立三重和Rogers--Ramanujan型恒等式，证实了Mizuno提出的除两个外的所有指标为$(1,2,2)$的三阶Nahm和的模性，并证明了两个向量值函数的模变换公式。
Triples and quadruples of consecutive squares or non-squares in a finite field
本文研究有限域中连续平方数或非平方数的三元组与四元组计数，利用 Jacobsthal 和求值给出显式公式，推广了 Monzingo 的结果，并处理了 5 的幂阶情形。
A general method to express in terms of Gauss sums the number of rational points of subschemes of projective schemes over finite fields is applied to the image of the triple embedding $\mathbb{P}^1\hookrightarrow\mathbb{P}^3$. As a consequence, we obtain a non-trivial description of the value of a Kloosterman-sum-like exponential sum.
Hypergeometric Motives from Toric Hypersurfaces
本文研究由 d 维环面中 d+2 个单项式线性组合定义的一般超曲面的两种紧化，证明其 F_q-点数量可由有限超几何和表示，并通过引入 gamma 三元组将有限超几何和推广到素数幂情形，推广了 Beukers、Cohen 和 Mellit 的工作。
Let $r\geq 1$ be a positive integer, $A$ a real positive semi-definite symmetric $r\times r$ rational matrix, $B$ a rational vector of length $r$, and $C$ a rational scalar. Nahm's problem is to find all triples $(A,B,C)$ such that the $r$-fold $q$-hypergeometric series $$f_{A,B,C}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T}\in (\mathbb{Z}_{\geq 0})^r} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}}$$ becomes a modular form, and we call such $(A,B,C)$ a modular triple. When the rank $r=3$, after extensive computer searches, Zagier provided twelve sets of conjec…
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。
The $α$-representation for Tait coloring and sums over spanning trees
本文提出极大平面图 Tait 染色数的 $α$-表示，通过面赋值构造边权重，利用生成树求和与 Legendre 符号定义 $w_G$，证明 Tait 染色数等于满足奇顶点条件的 $w_G$ 的三倍求和。
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