Chan and Kumchev studied averages of the first and second moments of Ramanujan sums. In this article, we extend this investigation by estimating the higher moments of averages of Ramanujan sums using the Brèteche Tauberian theorem. We also give a result for the moments of averages of Cohen-Ramanujan sums.
Ramanujan sums have been studied and generalized by several authors. For example, Nowak studied these sums over quadratic number fields, and Grytczuk defined that on semigroups. In this note, we deduce some properties on sums of generalized Ramanujan sums and give examples on number fields. In particular, we have a relational expression between Ramanujan sums and residues of Dedekind zeta functions.
Let $c_q(n)$ be the Ramanujan sums and let $S(m,n;q)$ be the Kloosterman sums. We study the eigenvalues of $q \times q$ matrices whose $(m,n)$ entry is $c_q (m-n)$ or $S(m,n;q)$ where $q$ is a fixed positive integer. We also study the eigenvalues of matrices whose entries are sums of Ramanujan sums or sums of Kloosterman sums.
Let $c_q(n)$ denote the Ramanujan sum modulo $q$, and let $x$ and $y$ be large reals, with $x = o(y)$. We obtain asymptotic formulas for the sums $$\sum_{n \le y}(\sum_{q \le x} c_q(n))^k \qquad (k = 1, 2).$$
The Ramanujan sum $c_n(k)$ is defined as the sum of $k$-th powers of the primitive $n$-th roots of unity. We investigate arithmetic functions of $r$ variables defined as certain sums of the products $c_{m_1}(g_1(k))...c_{m_r}(g_r(k))$, where $g_1,..., g_r$ are polynomials with integer coefficients. A modified orthogonality relation of the Ramanujan sums is also derived.
We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also construct a new Ramanujan expansion for the divisor function. While our expansion is amenable to a continuous and absolutely convergent real variable generalization, it only interpolates the divisor function locally on $\mathbb{R}$.
We use a specialization of Ramanujan's ${}_1ψ_1$ summation to give a new proof of a recent formula of Hickerson and Mortenson which expands a special family of Hecke-type double sums in terms of Appell-Lerch sums and theta functions.
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 derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Toth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose a generalization of Ramanujan sum given by E. Cohen and derive the weighted averages corresponding to the versions of the weighted averages established by Toth.
We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd's, the Gamma function, the Bernoulli polynomials, and binomial coefficients.
For a number field $\mathbb{K}$, and integral ideals $\mathcal{I}$ and $\mathcal{J}$ in its number ring $\mathcal{O}_{\mathbb{K}}$, Nowak studied the asymptotic behaviour of the average of Ramanujan sums $C_{\mathcal{J}}({\mathcal{I}})$ over both ideals $\mathcal{I}$ and $\mathcal{J}$. In this article, we extend this investigation by establishing asymptotic formulas for the second moment of averages of Ramanujan sums over quadratic and cubic number fields, thereby generalizing previous works of Chen, Kumchev, Robles, and Roy on moments of averages of Ramanujan sums over rationals. Additionally…
In this paper we study Ramanujan sums $c_{\bf m}(\bf n)$, where $ {\bf m}$ and ${\bf n}$ are integral ideals in an arbitrary quadratic number field. We give some new results about the asymptotic behavior of sums of $c_{\bf m}(\bf n)$ over both $ {\bf m}$ and $ {\bf n}$.
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvy's paper \cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
The polynomial Ramanujan sum was first introduced by Carlitz [7], and a generalized version by Cohen [10]. In this paper, we study the arithmetical and analytic properties of these sums, derive various fundamental identities, such as H older formula, reciprocity formula, orthogonality relation and Davenport{Hasse type formula. In particular, we show that the special Dirichlet series involving the polynomial Ramanujan sums are, indeed, the entire functions on the whole complex plane, we also give a square mean values estimation. The main results of this paper are new appearance to us, which ind…
We prove a simple formula for the main value of $r$-even functions and give applications of it. Considering the generalized Ramanujan sums $c_A(n,r)$ involving regular systems $A$ of divisors we show that it is not possible to develop a Fourier theory with respect to $c_A(n,r)$, like in the the usual case of classical Ramanujan sums $c(n,r)$.
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.
In the present paper, we introduce a multiple Ramanujan sum for arithmetic functions, which gives a multivariable extension of the generalized Ramanujan sum studied by D. R. Anderson and T. M. Apostol. We then find fundamental arithmetic properties of the multiple Ramanujan sum and study several types of Dirichlet series involving the multiple Ramanujan sum. As an application, we evaluate higher-dimensional determinants of higher-dimensional matrices, the entries of which are given by values of the multiple Ramanujan sum.
Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define the generalized Ramanujan sums over $K$ and show some properties.
We introduce new analogues of the Ramanujan sums, denoted by $\widetilde{c}_q(n)$, associated with unitary divisors, and obtain results concerning the expansions of arithmetic functions of several variables with respect to the sums $\widetilde{c}_q(n)$. We apply these results to certain functions associated with $σ^*(n)$ and $ϕ^*(n)$, representing the unitary sigma function and unitary phi function, respectively.
In this paper, we consider a general form of the analogue of Ramanujan's sum in the ring of polynomials over a finite field. We first prove some multiplicative properties of such functions before considering their finite Fourier series and some specific examples. In the end, we also prove a result about the Dirichlet series of such functions.
For two arithmetical functions $f$ and $g$, we study the convolution sum of the form $\sum_{n \le N} f(n) g(n+h)$ in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan expansion of an arithmetical function and extend our earlier works in this setup.
We continue our study of convolution sums of two arithmetical functions $f$ and $g$, of the form $\sum_{n \le N} f(n) g(n+h)$, in the context of heuristic asymptotic formulæ. Here, the integer $h\ge 0$ is called, as usual, the {\it shift} of the convolution sum. We deepen the study of finite Ramanujan expansions of general $f,g$ for the purpose of studying their convolution sum. Also, we introduce another kind of Ramanujan expansion for the convolution sum of $f$ and $g$, namely in terms of its shift $h$ and we compare this \lq \lq shifted Ramanujan expansion\rq \rq, with our previous finite e…
Let K be a cubic number field. In this paper, we study the Ramanujan sums c_{J}(I), where I and J are integral ideals in O_{K}. The asymptotic behaviour of sums of c_{J}(I) over both I and J is investigated.
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…
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…
Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in multi-variables over $\mathbb{A}_{+}$ can be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums. These are analogues of classical results over $\mathbb{N}$ by Winter, Delange and Tóth.
If $f(z)$ is a modular form of weight $k$, then the differential operator $\vartheta_k$ defined by $\vartheta_k(f) = \frac{1}{2πi} \frac{d}{dz}f(z) - \frac{k}{12} E_2(z) f(z)$ (known as the Ramanujan-Serre derivative map) is a modular form of weight $k+2$. In this paper, we obtain a simple extension of this map and use it to get a general method to derive certain convolution sums of the divisor functions (using the theory of modular forms). Explicit expressions are given for four types of convolution sums and we provide many examples for all these types of sums.
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…
In this paper, we prove supercongruence relations for truncated $N$-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogs of some Ramanujan-type supercongruences.
The theory of supercharacters, recently developed by Diaconis-Isaacs and Andre, can be used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.
The Ramanujan sum c_n(k) and a_n(k), the kth coefficient of the nth cyclotomic polynomial, are completely symmetric expressions in terms of primitive nth roots of unity. For fixed k we study the value distribution of c_n(k) (following A. Wintner) and a_n(k) (partly following H. Moller). In particular we disprove a 1970 conjecture of H. Moller on the average (over n) of a_n(k). We show that certain symmetric functions in primitive roots considered by the Dence brothers are related to the behaviour of c_{p-1}(k) and a_{p-1}(k) as p ranges over the primes and study their value distribution as wel…
Ramanujan sums are exponential sums with exponent defined over the irreducible fractions. Until now, they have been used to provide converging expansions to some arithmetical functions appearing in the context of number theory. In this paper, we provide an application of Ramanujan sum expansions to periodic, quasiperiodic and complex time series, as a vital alternative to the Fourier transform. The Ramanujan-Fourier spectrum of the Dow Jones index over 13 years and of the coronal index of solar activity over 69 years are taken as illustrative examples. Distinct long periods may be discriminate…
A highly strong upper estimate in the modified asymptotic formula for sums of the primes' reciprocals is proved to be necessary (as well as sufficient) in order the Ramanujan inequality holds true. Some other criteria in similar terms are also obtained.
We study the representation theory of a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of "magical" combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.
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 $q$.
Generalization of Ramanujan's formula for sums of half-integer powers of consecutive integers via formal Bernoulli series
The George Washington University（乔治华盛顿大学）；Lehman College of the City University of New York（纽约市立大学莱曼学院）；Icahn School of Medicine at Mount Sinai（西奈山伊坎医学院）；Medgar Evers College of the City University of New York（纽约市立大学梅加尔埃弗斯学院）
Galois-Orbit Structure, Ramanujan Sums, and Stable-Range Collapse for Cyclotomic Cosine Formulas
Departamento de Matemáticas, Sede Medellín, Universidad Nacional de Colombia（哥伦比亚国立大学麦德林校区数学系）；Departamento de Matemáticas, Universidad EAFIT（伊法蒂大学数学系）
We prove that $q+1$-regular Morgenstern Ramanujan graphs $X^{q,g}$ (depending on $g\in\mathbb{F}_q[t]$) have diameter at most $\left(\frac{4}{3}+\varepsilon\right)\log_{q}|X^{q,g}|+O_{\varepsilon}(1)$ (at least for odd $q$ and irreducible $g$) provided that a twisted Linnik-Selberg conjecture over $\mathbb{F}_q(t)$ is true. This would break the 30 year-old upper bound of $2\log_{q}|X^{q,g}|+O(1)$, a consequence of a well-known upper bound on the diameter of regular Ramanujan graphs proved by Lubotzky, Phillips, and Sarnak using the Ramanujan bound on Fourier coefficients of modular forms. We a…
In a 1916 paper, Ramanujan studied the additive convolution $S_{a, b}(n)$ of sum-of-divisors functions $σ_a(n)$ and $σ_b(n)$, and proved an asymptotic formula for it when $a$ and $b$ are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real $a$ and $b$. Ramanujan's conjecture was subsequently proved by Ingham, and then by Halberstam with a power saving error term. In this paper, we give a new proof of Ramanujan's conjecture that obtains lower order terms in the asymptotics for most ranges of the parameters. We also describe a connection to a c…
In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and it is possible that Ramanujan derived these identities with the intent of attacking these famous problems. Similar famous unsolved problems are connected with $f_K(n)$, the number of integral ideals of norm $n$ in an algebraic number field $K$. In this paper we establish Riesz sum identities containing an "extra" parameter and involving $f_K(n)$, or divisor f…
In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…
We highlight modular equations discovered by Somos and Ramanujan, and use them to prove new relations between lattice sums and hypergeometric functions. We also discuss progress towards solving Boyd's Mahler measure conjectures, and we conjecture a new formula for $L(E,2)$ of conductor 17 elliptic curves.
