Ramanujan Sums Research Articles

A conjectural extension of Hecke\u2019s converse theorem

We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.

On the polynomial Ramanujan sums over finite fields

The polynomial Ramanujan sum was first introduced by Carlitz (Duke Math J 14:1105–1120, 1947), and a generalized version by Cohen (Duke Math J 16:85–90, 1949). In this paper, we study the arithmetical and analytic properties of these sums, deriving various fundamental identities, such as Holder 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, and we also give a square mean values estimation. The main results of this paper are new appearance to us, which indicate the particularity of the polynomial Ramanujan sums.

Ramanujan expansions of arithmetic functions of several variables

We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel.

Unweighted linear congruences with distinct coordinates and the Varshamov–Tenengolts codes

In this paper, we first give explicit formulas for the number of solutions of unweighted linear congruences with distinct coordinates. Our main tools are properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions. Then, as an application, we derive an explicit formula for the number of codewords in the Varshamov--Tenengolts code $VT_b(n)$ with Hamming weight $k$, that is, with exactly $k$ $1$'s. The Varshamov--Tenengolts codes are an important class of codes that are capable of correcting asymmetric errors on a $Z$-channel. As another application, we derive Ginzburg's formula for the number of codewords in $VT_b(n)$, that is, $|VT_b(n)|$. We even go further and discuss connections to several other combinatorial problems, some of which have appeared in seemingly unrelated contexts. This provides a general framework and gives new insight into all these problems which might lead to further work.

Two-Dimensional Period Estimation by Ramanujan's Sum

Period estimation in one dimension (1-D) has been studied for years. However, 2-D period estimation is still a hard problem since it has three parameters to determine: length, width, and direction. Recently, a special kind of 2-D function called 2D-gcd-delta function is proposed. It has close relationship with Ramanujan's sum and the 2-D periodicity matrix. In this paper, we will describe how to use this function to decompose an image into subband signals. Each subband signal will have its own periodicity matrix so we also provide a simple algorithm to calculate the least common multiple (LCM) of those subband signal periodicity matrices. Concrete experiments are given to prove the robustness of the proposed 2-D period estimation method.

Sums of averages of generalized Ramanujan sums

For any positive integers k and j, we consider some asymptotic formulas for weighted averages of the Cohen–Ramanujan sums sk(s)(j)=∑d|k,ds|jf(d)g(kd) with any fixed positive integer s≥1, and any arithmetical functions f and g.

On Sums of Averages of Generalized Ramanujan Sums

We shall consider some formulas for weighted averages of the generalized Ramanujan sum $\sum_{d|\gcd(k,n)}f(d)g\left({k}/{d}\right)h\left({n}/{d}\right)$ for any arithmetical functions $f, g$ and $h$, with the weights concerning completely multiplicative functions, completely additive functions and others.

Closed-Form Orthogonal Ramanujan Integer Basis

In this letter, a closed-form orthogonal Ramanujan integer basis is proposed and obtained by performing Gram–Schmidt process from the Ramanujan sum and its circular shift. It has a surprisingly simple and sparse form, which is better than the original complete Ramanujan basis. The relationship between the original basis and proposed basis is also clearly illustrated. Therefore, the original basis can be replaced easily.

On a restricted linear congruence

Let [Formula: see text], [Formula: see text], and [Formula: see text] be all positive divisors of [Formula: see text]. For [Formula: see text], define [Formula: see text]. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence [Formula: see text], with [Formula: see text], [Formula: see text], is [Formula: see text] where [Formula: see text] is a Ramanujan sum. Some special cases and other forms of this problem have been already studied by several authors. The problem has recently found very interesting applications in number theory, combinatorics, computer science, and cryptography. The above explicit formula generalizes the main results of several papers, for example, the main result of the paper by Sander and Sander [J. Number Theory 133 (2013) 705–718], one of the main results of the paper by Sander [J. Number Theory 129 (2009) 2260–2266], and also gives an equivalent formula for the main result of the paper by Sun and Yang [Int. J. Number Theory 10 (2014) 1355–1363].

Certain weighted averages of generalized Ramanujan sums

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.

Restricted linear congruences

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence a1x1+⋯+akxk≡b(modn), with gcd⁡(xi,n)=ti (1≤i≤k), where a1,t1,…,ak,tk,b,n (n≥1) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of ai=ti=1(1≤i≤k). Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [Duke Math. J. 39 (1972) 521–527] gave a nice explicit formula for the number of such solutions when (a1,…,ak)=ti=1(1≤i≤k). The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.

Counting surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group with motivations from string theory and QFT

Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch et al. (2013) [12] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an explicit and practical formula for the number of such epimorphisms. As a consequence, we obtain an ‘equivalent’ form of Harvey's famous theorem on the cyclic groups of automorphisms of compact Riemann surfaces. Our main tool is an explicit formula for the number of solutions of restricted linear congruence recently proved by Bibak et al. using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions.

On shifted Mascheroni series and hyperharmonic numbers

TextIn this article, we study the nature of the forward shifted series σr=∑n>r|bn|n−r where r is a positive integer and bn are Bernoulli numbers of the second kind, expressing them in terms of the derivatives ζ′(−k) of zeta at the negative integers and Euler's constant γ. These expressions may be inverted to produce new series expansions for the quotient ζ(2k+1)/ζ(2k). Motivated by a theoretical interpretation of these series in terms of Ramanujan summation, we give an explicit formula for the Ramanujan sum of hyperharmonic numbers as an application of our results. VideoFor a video summary of this paper, please visit https://youtu.be/uyLmgDh9JVs.

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.

The SIAM 100-Digit Challenge: A Decade Later

In 2002, L.N. Trefethen of Oxford University challenged the scientific community by ten intriguing mathematical problems to be solved, numerically, to ten digit accuracy each (I was one of the successful contestants; in 2004, jointly with three others of them, I published a book—Bornemann et al.: The SIAM 100-Digit Challenge, SIAM, Philadelphia, 2004—on the manifold ways of solving those problems). In this paper, I collect some new and noteworthy insights and developments that have evolved around those problems in the last decade or so. In the course of my tales, I will touch mathematical topics as diverse as divergent series, Ramanujan summation, low-rank approximations of functions, hybrid numeric-symbolic computation, singular moduli, self-avoiding random walks, the Riemann prime counting function, and winding numbers of planar Brownian motion. As was already the intention of the book, I hope to encourage the reader to take a broad view of mathematics, since one lasting moral of Trefethen’s contest is that overspecialization will provide too narrow a view for one with a serious interest in computation.

Explicit formulas of a generalized Ramanujan sum

In this paper, explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.

Intrinsic Integer-Periodic Functions for Discrete Periodicity Detection

In this letter we focus on discrete integer-periodic signals, whose periodicity are different from the periodicity of continuous periodic signals in many aspects. We introduce a class of discrete periodic signals called intrinsic integer-periodic function (IIPF). An IIPF contains only a single period in terms of downsampling, which leads to some interesting properties for analyzing periodic components from a discrete signal. We show that one can use Ramanujan sum to decompose discrete periodic signals into IIPF components. Finally, we also propose an integer periodic spectrum rather than frequency spectrum. Our results show that the proposed integer periodic spectrum outperforms the conventional Ramanujan Fourier transform.

On sums of generalized Ramanujan sums

Ramanujan sums have been studied and generalized by several authors. For example, Nowak [8] studied these sums over quadratic number fields, and Grytczuk [4] 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.

Ramanujan Sums in the Context of Signal Processing—Part II: FIR Representations and Applications

The mathematician Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). In a companion paper (Part I), properties of Ramanujan sums were reviewed, and Ramanujan subspaces S_q introduced, of which the Ramanujan sum is a member. In this paper, the problem of representing finite duration (FIR) signals based on Ramanujan sums and spaces is considered. First, it is shown that the traditional way to solve for the expansion coefficients in the Ramanujan-sum expansion does not work in the FIR case. Two solutions are then developed. The first one is based on a linear combination of the first N Ramanujan-sums (with N being the length of the signal). The second solution is based on Ramanujan subspaces. With q_1, q_2,..., q_K denoting the divisors of N; it is shown that x(n) can be written as a sum of K signals x_(qi) (n) ∈ S_(qi). Furthermore, the i_(th) signal x_(qi) (n) has period q_i, and any pair of these periodic components is orthogonal. The components x_(qi) (n) can be calculated as orthogonal projections of x(n) onto Ramanujan spaces S_(qi). Then, the Ramanujan Periodic Transform (RPT) is defined based on this, and is useful to identify hidden periodicities. It is shown that the projection matrices (which compute x_(qi) (n) from x(n)) are integer matrices except for an overall scale factor. The calculation of projections is therefore rendered easy. To estimate internal periods N_∞ <; N of x(n), one only needs to know which projection energies are nonzero.

Ramanujan Sums in the Context of Signal Processing—Part I: Fundamentals

The famous mathematician S. Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). For any fixed integer q, this is a sequence in n with periodicity q. Ramanujan showed that many standard arithmetic functions in the theory of numbers, such as Euler's totient function φ(n) and the Mobius function μ(n), can be expressed as linear combinations of c_q(n), 1 ≤ q ≤ ∞. In the last ten years, Ramanujan sums have aroused some interest in signal processing. There is evidence that these sums can be used to extract periodic components in discrete-time signals. The purpose of this paper and the companion paper (Part II) is to develop this theory in detail. After a brief review of the properties of Ramanujan sums, the paper introduces a subspace called the Ramanujan subspace S_q and studies its properties in detail. For fixed q, the subspace S_q includes an entire family of signals with properties similar to c_q(n). These subspaces have a simple integer basis defined in terms of the Ramanujan sum c_q(n) and its circular shifts. The projection of arbitrary signals onto these subspaces can be calculated using only integer operations. Linear combinations of signals belonging to two or more such subspaces follows certain specific periodicity patterns, which makes it easy to identify periods. In the companion paper (Part II), it is shown that arbitrary finite duration signals can be decomposed into a finite sum of orthogonal projections onto Ramanujan subspaces.