László Tóth
We investigate the polynomials $\sum_{k=0}^{n-1} c_n(k)x^k$ and $\sum_{k=0}^{n-1} |c_n(k)| x^k$, where $c_n(k)$ denote the Ramanujan sums. We point out connections and analogies to the cyclotomic polynomials.
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Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients
第n阶分圆多项式的第k个系数的Ramanujan和c_n（k）与a_n（k）是关于第n阶本原单位根的完全对称表达式。对于固定的k，我们研究了c_n（k）的值分布（继A. Wintner）和a_n（k）（部分遵循H. Moller）。特别是我们反驳了1970年的猜想H。Moller关于a_n（k）的平均（在n上）。我们证明了Dence兄弟所考虑的原根中的某些对称函数与c_{p-1}（k）和a_{p-1}（k）在素数上的p范围的行为有关，并研究了它们的值分布。本文是第二作者的硕士论文项目的产物，在第一作者的监督下在Korteweg-de Vries研究所（阿姆斯特丹大学）进行，并以硕士论文风格撰写。我们感谢Yves Gallot提供的数字援助。
Constancy of an Infinite Cyclotomic Product via Ramanujan Sums
本文通过拉马努金求和证明了无限循环多项式乘积在单位圆内的恒定性，并揭示了相关恒等式对无限乘积的进一步影响。
Galois-Orbit Structure, Ramanujan Sums, and Stable-Range Collapse for Cyclotomic Cosine Formulas
本文结合伽罗瓦轨道与有限频率展开，研究分圆余弦求值，揭示值层面伽罗瓦不变性缺乏稳定范围刚性，并证明特定模式族在足够大水平下坍缩为仿射函数，最终用拉马努金和给出精确轨道公式。
Divisibility criteria and coefficient formulas for cyclotomic polynomials
本文给出多项式被分圆多项式整除的充要条件，推导拉马努金和的新公式，并揭示分圆多项式系数的新性质及指数相差素因子时的递推关系。
Moments of Averages of Ramanujan Sums over Number Fields
本文在广义 Lindelöf 假设下给出了任意数域上 Ramanujan 和第一矩和第二矩均值的渐近公式，并在无条件下估计了分圆数域上 Ramanujan 和的第二矩。
Some integrals of the Glaisher-Ramanujan type are established in a more general form than in previous studies. As an application we prove some Ramanujan-type series identities, as well as a new formula for the Dirichlet beta function at the value $s=3.$
Remarks on a Formula of Ramanujan
在Mertens猜想平均形式和黎曼zeta函数零点纵坐标有理线性无关的假设下，分析了拉马努金著名公式各项的精细结构。
Remarks on generalized Ramanujan sums and even functions
本文证明了$r$ -偶函数的主值的一个简单公式，并给出了它的应用.考虑到广义Ramanujan和$c_A(n,r)$涉及正规系统$A$，我们表明，它是不可能发展的傅立叶理论关于$c_A(n,r)$，像在通常情况下的经典Ramanujan和$c(n,r)$ .
We observe that five polynomial families have all of their zeros on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, Smyth and Wang. Our proofs rely upon theorems of Schinzel, and Lakatos and Losonczi and some generalizations.
Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a "divergent" Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a supercongruence recently proved by Wang: for any prime $p>3$, $$ \sum_{k=0}^{p-1} (3k-1)\frac{(\frac{1}{2})_k (-\frac{1}{2})_k^2 }{k!^3}4^k\equiv p-2p^3 \pmod{p^4}, $$ where $(a)_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer symbol.
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 $q$.
By applying Chinese remainder theorem for coprime polynomials and the "creative microscoping" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. The original $q$-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to $1$ (l'Hôpital's rule is utilized here). In particular, we prove a complete $q$-analogue of the (J.2) supercongruence of Van Hamme and a complete $q$-analogue of a "divergent" Ramanujan-type supercon…
The $n^{th}$ cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers $n$. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their appearance, sometimes even at the same time! In this paper we present a survey of these formulas which until now were scattered in the literature and introduce an unified approach…
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude…
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…
A remark on modular equations involving Rogers-Ramanujan continued fraction via $5$-dissections
本文通过研究Ramanujan theta函数的5-剖分并建立与Rogers-Ramanujan连分数相关的恒等式，重新证明了Ramanujan遗失笔记本中的一个模方程，并进一步建立了涉及$R(q)$至$R(q^{16})$的新模方程。
On a Special Metric in Cyclotomic Fields
本研究在奇素数$p$对应的分圆域上引入一种具有伽罗瓦不变性的特殊度量，证明其整数环内原点中心超立方体的点对距离在$p$、$N$足够大时几乎均等，归一化后为$1/\sqrt{6}$。
Remarks on Hilbert's tenth problem and the Iwasawa theory of elliptic curves
本文利用岩泽理论给出椭圆曲线在分圆塔中秩为常数的显式条件，并探讨其在数环希尔伯特第十问题上的潜在应用。
An iterative approach toward hypergeometric accelerations
本研究受Chu的超几何加速思路启发，基于Wilf依赖Zeilberger算法导出非齐次差分方程的加速方法，提出迭代拓展方法，证明了多个通用常数的加速拉马努金型公式及相关新加速结果。
A natural variant of Lehmer's conjecture that the Ramanujan $τ$-function never vanishes asks whether, for any given integer $α$, there exist any $n \in \mathbb{Z}^+$ such that $τ(n) = α$. A series of recent papers excludes many integers as possible values of the $τ$-function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $τ(n)$. We synthesize these results and methods to prove that if $0 < |α| < 100$ and $α\notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$, then $τ(n) \neq α$ for all $n > 1$. Moreover, if $α\in T$ and $τ(n) =…
An integer $k$ is called regular (mod $n$) if there exists an integer $x$ such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$ possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that $k^{m+1} \equiv k$ (mod $n$). Let $\varrho(n)$ denote the number of regular integers (mod $n$) in the set $\{1,2,\ldots,n\}$. This is an analogue of Euler's $ϕ$ function. We introduce the multidimensional generalization of $\varrho$, which is the analogue of Jordan's function. We establish identities for the power sums of regular integers (mod $n$) and for some other finite…
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 establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori-Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bound…
Remarks on Greenberg's conjecture for Galois representations associated to elliptic curves
本文研究椭圆曲线 Galois 表示的 Greenberg 猜想，借助 Coates–Sujatha 精细 Selmer 群定理，在若干 Galois 理论条件下证明可约情形结论，并讨论不可约情形的 μ 不变量消失。
Combinatorial Aspects of Elliptic Curves
给定一条椭圆曲线C，本文研究有限域F_{q^k}上C的点数$N_k = #C(F_{q^k})$.当k遍历正整数时，这个数列除了通常的数论解释外，还具有许多显著的组合性质。特别地，我们证明了$N_k = - W_k(q, - N_1)$，其中W_k（q，t）是轮图的生成树个数的（q，t）-模拟.此外，我们开发了一个行列式公式N_k的特征值可以明确地写在q，N_1，和单位根。本文还讨论了与N_k的因式分解有关的二元多项式的一个新序列，由于它的各种性质，我们称之为椭圆分圆多项式。
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