Gert Almkvist
An asymptotic formula for the number of partitions into p-cores is derived. As a byproduct some integer valued trigonometric sums are found
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting $q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper's $q$-trigonometric identities.
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of the authors. Reciprocity theorems for certain trigonometric sums are also established.
Two classes of finite trigonometric sums, each involving only $\sin$'s, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and modular equations.
We show that identities involving trigonometric sums recently proved by Harshitha, Vasuki and Yathirajsharma, using Ramanujan's theory of theta functions, were either already in the literature or can be proved easily by adapting results that can be found in the literature. Also we prove two conjectures given in that paper. After mentioning many other works dealing with identities for various trigonometric sums, we end this paper by describing an automated approach for proving such trigonometric identities.
Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated with primitive odd characters. In this paper, we establish two general theorems involving both sines and cosines, with more attention given to cosine sums in the several examples that we provide.
We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle.
Evaluations and relations for finite trigonometric sums
该研究针对有限三角和的求值问题，建立三角和的互反及三个和式关系式，定义带互素条件的Ramanujan和类似物并求值，得到与黎曼假设相关的判别准则。
We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were systematically first studied by Berndt and Zaharescu; their proofs involved complex contour integration. We show how to prove most of Berndt-Zaharescu's and some new identities by elementary methods of discrete Fourier Analysis.
We introduce some new higher dimensional generalizations of the Dedekind sums associated with the Bernoulli functions and of those Hardy sums which are defined by the sawtooth function. We generalize a variant of Parseval's formula for the discrete Fourier transform to derive finite trigonometric representations for these sums in a simple unified manner. We also consider a related sum involving the Hurwitz zeta function.
Finite trigonometric sums appear in various branches of Physics, Mathematics and their applications. For p; q to coprime positive integers and r we consider the finite trigonometric sums involving the product of three trigonometric functions.
We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}γ(y)\exp(2πi a λ^{xy}/p)\Big|, \end{eqnarray*} where $λ$ is an element of the residue ring modulo a large prime number $p,$ $\mathcal{X}$ and $\mathcal{Y}$ are arbitrary subsets of the residue ring modulo $p-1$ and $γ(n)$ are any complex numbers with $|γ(n)| \le 1.$ In particular, we improve several previously known bounds.
We obtain new trigonometric identities, which are some product-to-sum type formulas for the higher derivative of the cotangent and cosecant functions. Further, from specializations of our formulas, we derive not only various known reciprocity laws of generalized Dedekind sums but also new reciprocity laws of generalized Dedekind sums.
Trigonometric analogue of the identities associated with twisted sums of divisor functions
本文受 Ramanujan 丢失笔记启发，推导出与除数函数扭曲和相关恒等式的三角函数等价版本，并给出了 $r_6(n)$ 的模拟恒等式。
We obtain new trigonometric identities, which are product-to-sum type formulas for derivative of the cosecant and cotangent functions. Further, from specializations of our formulas, we derive new reciprocity laws of generalized Dedekind sums.
Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2π/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2π/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bi…
Ramanujan's trigonometric sum $c_q(n)$ can be interpreted as a set of trigonometric moments of a finite measure concentrated at primitive $q$-th roots of unity with equal masses. This gives rise to sets of corresponding polynomials orthogonal on the unit circle. We present explicit expressions of these polynomials for special values of $q$, e.g. when $q=p$ or $q=2p$ or $q=p^k$, where $p$ is a prime number. We generalize this procedure taking the Kronecker polynomial instead of cyclotomic one. In this case the moments are expressed as finite sums of $c_q(n)$ with different $q$.
Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions. These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From …
We examine the four Féjer-type trigonometric sums of the form \[S_n(x)=\sum_{k=1}^n \frac{f(g(kx))}{k}\qquad (0<x<π)\] where $f(x)$, $g(x)$ are chosen to be either $\sin x$ or $\cos x$. The analysis of the sums with $f(x)=g(x)=\cos x$, $f(x)=\cos x$, $g(x)=\sin x$ and $f(x)=\sin x$, $g(x)=\cos x$ is reasonably straightforward. It is shown that these sums exhibit unbounded growth as $n\to\infty$ and also present `spikes' in their graphs at certain $x$ values for which we give an explanation. The main effort is devoted to the case $f(x)=g(x)=\sin x$, where we present arguments that strongly supp…
Relations between Chebyshev, Fibonacci and Lucas polynomials via trigonometric sums
本文通过有限三角和建立斐波那契、卢卡斯与切比雪夫多项式间的新恒等式，并给出其物理解释及与图有效电阻的联系。
The trigonometric polynomial on sums of two squares, an additive problem and generalisation
本文研究两互素平方和奇数集合上的三角多项式，证明其在有理点附近的强估计，并结合受转移原理启发的圆法解决三类正密度子集元素相加的三表示问题。
Moment generating functions and moderate deviation principles for lacunary trigonometric sums
本文研究缺项三角和的矩生成函数，确定算术效应起作用的精确阈值，证明其中偏差原理，证实尾概率在中心极限定理与大偏差原理区间内符合高斯行为。
We present a new and elegant integral approach to computing the Gardner-Fisher trigonometric power sum, which is given by $$ S_{m,v}=\left(\frac π{2m}\right)^{2v}\sum_{k=1}^{m-1}\cos^{-2v}\left(\frac{kπ}{2m}\right)\, , $$ We present a new and elegant integral approach to computing the Gardner-Fisher trigonometric power sum, which is given by $$ S_{m,v}=\left(\frac π{2m}\right)^{2v}\sum_{k=1}^{m-1}\cos^{-2v}\left(\frac{kπ}{2m}\right)\, , $$ where $m$ and $v$ are positive integers. This method not only confirms the results obtained earlier by an empirical method, but it is also much more expedie…
In this paper we construct trigonometric functions of the sum T_{p}(h,k), which is called Dedekind type DC-(Dahee and Changhee) sums. We establish analytic properties of this sum. We find trigonometric representations of this sum. We prove reciprocity theorem of this sums. Furthermore, we obtain relations between the Clausen functions, Polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums and the sum T_{p}(h,k). We also give some applications related to these sums and functions.
Moderate deviation principles and Mod-Gaussian convergence for lacunary trigonometric sums
本文研究缺项三角和的中偏差原理与Mod-Gaussian收敛，引入相关图并采用累积量方法，证明√n与n/log(n)尺度间无算术效应，还借助不同工具得到mod-Gaussian收敛结果，为明确大偏差特殊现象的显现尺度提供研究基础。
We combine an extended version of Bailey's transform with an identity of Bressoud and with some identities of Berkovich and Warnaar to prove a variety of positivity results for alternating sums involving partition functions.
In this paper, we investigate two methods to express the natural powers of $2$ as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients as a sum over partitions. The second method invokes the central binomial coefficients and the logarithmic differentiation of their generating function. Some experimental results suggest the existence of other methods of decomposing the power of $2$ as sums over partitions.
Trigonometric Determinants via special values of Dirichlet $L$-Functions
本文通过狄利克雷 L 函数的特殊值研究三角行列式，并推导出正弦矩阵行列式的显式公式。
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
Contour Integrations and Parity Results of Cyclotomic Euler Sums and Multiple Polylogarithm Function
该研究通过级数定义扩展三角函数，用围道积分法研究分圆欧拉和及多重多对数函数的奇偶性，得到多类相关显式公式，并推导出经典及交错欧拉和奇偶性的已知公式。
An Algebraic Generalization of the Ramanujan Sum
本文通过多项式取余法推广拉马努金和，将其表示为带互质条件的有限三角和，还给出了带奇偶条件的Levenshtein码大小的显式公式，解决了SVT码的重要开放问题。
Improved asymptotics for moments of reciprocal sums for partitions into distinct parts
针对互异整数分拆中的倒数和与倒数平方和，本文改进了 $s_1(n)$ 和 $s_2(n)$ 的渐近估计，并处理了非模、非乘积生成函数带来的技术困难。
Modularity of moments of reciprocal sums for partitions into distinct parts
本文研究了互异分拆中部分倒数 $k$ 次幂之和的生成函数的模性，证明其与 Maass Eisenstein 级数和半调和 Maass 形式相关。
We study representations of integers as sums of the form $\pm a_1\pm a_2\pm \dotsb \pm a_n$, where $a_1,a_2,\ldots$ is a prescribed sequence of integers. Such a sequence is called an Erdős-Surányi sequence if every integer can be written in this form for some $n\in\mathbb{N}$ and choices of signs in infinitely many ways. We study the number of representations of a fixed integer, which can be written as a trigonometric integral, and obtain an asymptotic formula under a rather general scheme due to Roth and Szekeres. Our approach, which is based on Laplace's method for approximating integrals, c…
Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums
本文将广义 τ_k(n) 表示为二项式系数加权分拆和，从而建立其与 t-正则分拆函数 R_t(n) 及加权复合和之间的同余恒等式。
A note on the distribution of clusters and deserts of prime numbers
本文利用三角多项式值分布公式，研究加权 von Mangoldt 函数值之和的分布，并给出其服从类正态分布的证据。
Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions
本文研究强加性权函数下的加权分拆函数，在更广背景下建立其生成级数的渐近公式，并利用 Montgomery-Vaughan 经典结果估计小弧上带加性系数的指数和。
Exponential sums over Möbius convolutions with applications to partitions
本文研究了莫比乌斯卷积权重下的划分问题，推导了划分数量的上界及渐近公式，并给出了与黎曼zeta函数零点相关的显式公式。
Sums of the form $\sum_{q \leq N_1 < \cdots < N_m \leq n}{a_{(m);N_m}\cdots a_{(2);N_2}a_{(1);N_1}}$ date back to the sixteen century when Viète illustrated that the relation linking the roots and coefficients of a polynomial had this form. In more recent years, such sums have become increasingly used with a diversity of applications. In this paper, we develop formulae to help with manipulating such sums (which we will refer to as multiple sums). We develop variation formulae that express the variation of multiple sums in terms of lower order multiple sums. Additionally, we derive a set of par…
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。
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。
Exact formulae for ranks of partitions
本文通过证明Bringmann公式在特定条件下的精确性，为划分秩的精确公式提供了新方法，并推导出Dyson猜想的新证明途径。
We examine the $q$-Pell sequences and their applications to weighted partition theorems and values of $L$-functions. We also put them into perspective with sums of tails.
Series over Bessel functions as series in terms of Riemann's zeta function
本文利用Hurwitz公式和三角级数求和公式，将Bessel函数级数转化为Riemann zeta函数级数，并求得有限和。
Sums of the form $\sum_{N_m=q}^{n}{\cdots \sum_{N_1=q}^{N_2}{a_{(m);N_m}\cdots a_{(1);N_1}}}$ where the $a_{(k);N_k}$'s are same or distinct sequences appear quite often in mathematics. We will refer to them as recurrent sums. In this paper, we introduce a variety of formulas to help manipulate and work with this type of sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order $m$ to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order o…
Distribution of Alternating Sums of Parts in Partitions
本文研究分拆中交替和统计量的分布，计算其所有矩并用圆法证明，同时提出可规避传统方法复杂性、便于广泛推广的一般研究框架。
We study M(n,k,r), the number of orbits of {(a_1,...,a_k)\in Z_n^k | a_1+...+a_k = r (mod n)} under the action of S_k. Equivalently, M(n,k,r) sums the partition numbers of an arithmetic sequence: M(n,k,r) = sum_{t \geq 0} p(n-1,k,r+nt), where p(a,b,t) denotes the number of partitions of t into at most b parts, each of which is at most a. We derive closed formulas and various identities for such arithmetic partition sums. These results have already appeared in Elashvili/Jibladze/Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Alg. Combin. 10 (1999) 173-188, and the main resul…
We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $π$, and to connect sums over partitions to the Riemann zeta function, multiple zeta values, and other number-theoretic objects.
Let $A$ be a subset of positive integers. By $A$-partition of $n$ we understand the representation of $n$ as a sum of elements from the set $A$. For given $i, n\in\N$, by $c_{A}(i,n)$ we denote the number of $A$-partitions of $n$ with exactly $i$ parts. In the paper we obtain several result concerning sign behaviour of the sequence $S_{A,k}(n)=\sum_{i=0}^{n}(-1)^{i}i^{k}c_{A}(i,n)$, where $k\in\N$ is fixed. In particular, we prove that for a broad class $\cal{A}$ of subsets of $\N_{+}$ we have that for each $A\in \cal{A}$ we have $(-1)^{n}S_{A,k}(n)\geq 0$ for each $n, k\in\N$.
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…
Asymptotics of parity biases for partitions into distinct parts via Nahm sums
本文推广了Kim等人的奇偶偏差研究，利用圆法给出互异分拆的模N偏差渐近结果，同时提供了分析Nahm型q-超几何级数渐近性的独立有用技术。
We present a method of computing the generating function $f_P(\x)$ of $P$-partitions of a poset $P$. The idea is to introduce two kinds of transformations on posets and compute $f_P(\x)$ by recursively applying these transformations. As an application, we consider the partially ordinal sum $P_n$ of $n$ copies of a given poset, which generalizes both the direct sum and the ordinal sum. We show that the sequence $\{f_{P_n}(\x)\}_{n\ge 1}$ satisfies a finite system of recurrence relations with respect to $n$. We illustrate the method by several examples, including a kind of 3-rowed posets and the…
We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are $k$-gonal numbers; our proofs employ Ramanujan's theta functions. We explore applications to lacunary $q$-series, and to a new class of composition-theoretic Dirichlet series.
Monochromatic Sums and Products of Polynomials
本文证明了模式 $\{x,x+y,xy\}$ 在常数项为零且次数至少为一次的形式整数多项式空间上具有划分正则性及原始递归界，并为该模式在自然数上的划分正则性提供了首个原始递归界的新证明。
First we reprove two results in additive number theory due to Dombi and Chen & Wang, respectively, on the number of representations of n as the sum of two odious or evil numbers, using techniques from automata theory and logic. We also use this technique to prove a new result about the numbers represented by five summands. Furthermore, we prove some new results on the tenfold sums of the evil and odious numbers, as well as k-fold sums of similar sequences of integers, by using techniques of analytic number theory involving trigonometric sums associated with the (+-1)-characteristic sequences o…
Monochromatic sums and quotients in $\mathbb N$
本文证明了配置x,y,x+y,y/x在强无穷形式下的分拆正则性，扩展了Hindman定理，并研究了涉及x和y一次多项式乘积的分拆正则性问题。
Arithmetic sensitivity of cumulant growth in lacunary sums: transcendental versus algebraic ratio limits
本文研究稀疏三角求和的累积增长，发现其渐进行为对序列的算术结构敏感，超越数比极限情况下累积量线性增长，而代数比极限时可能呈现不同增长模式。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。