Arithmetical properties of multiple Ramanujan sums

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.     

Sums of products of Ramanujan sums

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 \({c_{m_1}(g_1(k))\cdots c_{m_r}(g_r(k))}\), where g ₁, . . . , g _r are polynomials with integer coefficients. A modified orthogonality relation of the Ramanujan sums is also derived.     

Distribution of averages of Ramanujan sums

The average value of a certain normalization of Ramanujan sums is determined in terms of Bernoulli numbers and odd values of the Riemann zeta function. The distribution of values and limiting behavior of such a normalization are then studied along subsets of Beurling type integers with positive density and sequences of moduli with constraints on the number of distinct prime factors.     

On an optimality property of Ramanujan sums

We evaluate , where the is taken over sequences satisfying . In particular we show that it is attained by taking for all , which reduces the summation over to a Ramanujan sum .

     

Certain weighted averages of generalized Ramanujan sums

We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Tóth 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 Tóth.     

Moments of averages of generalized Ramanujan sums

Let \(\beta \) be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by

$$\begin{aligned} c_{q,\beta }(n) := \sum \limits _{{{(h,{q^\beta })}_\beta } = 1} {{e^{2\pi inh/{q^\beta }}}}, \end{aligned}$$

where h ranges over the non-negative integers less than \(q^{\beta }\) such that h and \(q^{\beta }\) have no common \(\beta \)-th power divisors other than 1. The distribution of the average value of the Ramanujan sum is a subject of extensive research. In this paper, we study the distribution of the average value of \(c_{q,\beta }(n)\) by computing the k-th moments of the average value of \(c_{q,\beta }(n)\). In particular we have provided the first and second moments with improved error terms. We give more accurate results for the main terms than our predecessors. We also provide an asymptotic result for an extension of a divisor problem and for an extension of Ramanujan’s formula.

     

Averages of shifted convolution sums for arithmetic functions

Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(cm-2,…, c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|≤H |ΣX相似文献   

A generalization of regular convolutions and Ramanujan sums

The Ramanujan Journal - Regular convolutions of arithmetical functions were first defined by Narkiewicz (Colloq Math 10:81–94, 1963). Useful identities regarding generalizations of the...     

A note on character sums

We investigate certain character sums and prove some discrepancy-type inequalities for incomplete sums.      

A note on character sums

Let $\text{S(k) = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{p?1}}\text{(}\text{n}\text{p}\text{)n}{}^{\mathrm{k}}$ where p is a prime ≡ 3 mod 4 and k is an integer ≥ 3. Then S(k) frequently takes large values of each sign.     

On Ramanujan sums of a real variable and a new Ramanujan expansion for the divisor function

The Ramanujan Journal - 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...     

The average size of Ramanujan sums over quadratic number fields

This article is concerned with Ramanujan sums ${c_{\mathcal{I}_1}(\mathcal{I}),}$ where ${\mathcal{I},\mathcal{I}_1}$ are integral ideals in an arbitrary quadratic number field ${\mathbb{Q}(\sqrt{d}).}$ In particular, the asymptotic behavior of sums of ${c_{\mathcal{I}_1}(\mathcal{I}),}$ over both ${\mathcal{I}}$ and ${c_{\mathcal{I}_1}(\mathcal{I}),}$ is investigated.     

A short note on Dedekind sums

We will give a new proof for the fact that the values of Dedekind sums are dense on the real line.     

Classical Kloosterman sums: Representation theory, magic squares, and Ramanujan multigraphs

We consider 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.     

Approximating reals by sums of two rationals

We generalize Dirichlet's diophantine approximation theorem to approximating any real number α by a sum of two rational numbers with denominators 1?q₁,q₂?N. This turns out to be related to the congruence equation problem with 1?x,y?q1/2+?.     

A note on character sums with polynomial arguments

An improvement of Weil bound for a class of polynomials over GF(2ⁿ) is obtained.     

A note on sums of products of Bernoulli numbers

In this work we obtain a new approach to closed expressions for sums of products of Bernoulli numbers by using the relation of values at non-positive integers of the important representation of the multiple Hurwitz zeta function in terms of the Hurwitz zeta function.