Ramanujan Sums Research Articles

On the mean square average of special values of L-functions

Let χ be a Dirichlet character and L(s,χ) be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordanʼs and Eulerʼs totient function for the mean square average of L(1,χ) when χ ranges over all odd characters modulo k and L(2,χ) when χ ranges over all even characters modulo k. In principle, using our method, it is always possible to find the mean square average of L(r,χ) if χ and r⩾1 have the same parity and χ ranges over all odd (or even) characters modulo k, though the required calculations become formidable when r⩾3. Consequently, we see that for almost all odd characters modulo k, |L(1,χ)|<Φ(k), where Φ(x) is any function monotonically tending to infinity.

Coulomb gas partition function of a layered loop model

We consider a two-dimensional bi-layered loop model with a certain interlayer coupling and study its spectrum on a torus. Each layer consists of an O(n) model on a honeycomb lattice with periodic boundary conditions; these layers are stacked such that the links of the lattice intersect each other. A complex Boltzmann weight λ with unit modulus is assigned to each intersection of two loops each from each layer. The model is reduced to an inhomogeneous vertex model at a special point of parameters. The continuum partition function is represented, based on the idea of the Coulomb gas, by a path integral over two compact bosonic fields. The modular invariance of the partition function follows naturally. Further, because of the topological nature of the interlayer coupling, the fluctuation of loops decomposes into a local and a global part. The existence of the latter leads to a sum over all the pairs of torus knots, which can be Poisson resummed by the Möbius inversion formula. This reveals the operator content of the theory. The multiplicity of each operator is explicitly given by a combination of two Ramanujan sums. We calculate each scaling dimension as a function of λ. We present the flow of dimensions which connects the decoupled-O(1) models at λ = 1 and the layered-O(1) model with the non-trivial coupling λ = −1. The lower spectrum in the latter model is related to that of a known coset model.

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.

Ramanujan summation and the exponential generating function $\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta^{\prime}(-k)$

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 ∑n≥1log n enz 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 which follows a formula that connects the derivatives of the Riemann zeta-function at the negative integers to the Ramanujan sum of the divergent Euler sums ∑n≥1nkHn, k ≥ 0, where \(H_{n}=H_{n}^{(1)}\) . Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.

Doppler spectrum estimation by Ramanujan-Fourier Transform (RFT)

This work deals with the spectrum parameters estimation of a weather radar signal, namely velocity, variance and spectrum width. A new tool - based on the Ramanujan sums, cq(n), which come from the arithmetic functions of the number theory - has been introduced in signal processing. As compared to pulse pair and Fourier algorithms, new results have been obtained by the use of the Ramanujan-Fourier Transform (RFT) for the Doppler spectrum estimation of a real weather data provided by terminal pulse Doppler radar (WSR-88D).

Distribution of rational numbers in short intervals

A simplified proof for a well-distribution property for rational numbers is given and a connection with Riemann’s Hypothesis is pointed out. More precisely, we consider rational numbers with denominators of a given order of magnitude and show that the number of such numbers lying in a short interval of given length is normally close to its expectation in a mean square sense. The proof is elementary, using only Fourier series and Ramanujan sums. At the end of the paper, a variant of the circle method is discussed as an application.

On the Littlewood cyclotomic polynomials

In this article, we study the cyclotomic polynomials of degree N − 1 with coefficients restricted to the set { + 1 , − 1 } . By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p ( x ) with coefficients ±1 of even degree N − 1 is cyclotomic if and only if p ( x ) = ± Φ p 1 ( ± x ) Φ p 2 ( ± x p 1 ) ⋯ Φ p r ( ± x p 1 p 2 ⋯ p r − 1 ) , where N = p 1 p 2 ⋯ p r and the p i are primes, not necessarily distinct. Here Φ p ( x ) : = x p − 1 x − 1 is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree 2 α p β − 1 with odd prime p or separable polynomials of any odd degree.

HYBRID MEAN VALUE OF GENERALIZED BERNOULLI NUMBERS, GENERAL KLOOSTERMAN SUMS AND GAUSS SUMS

The main purpose of this paper is to use the properties of primitive characters, Gauss sums and Ramanujan's sum to study the hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums, and give two asymptotic formulae.

B-binomial convolution and associated arithmetical functions and their properties

In this paper we considered Apostol's article (2) and we defined for B-Binomial convolution associated arithmetical function S. We show that this function and B-Binomial convolution associated with Ramanujan Sums are connected.

Application of the Ramanujan Fourier Transform for the Analysis of Secondary Structure Content in Amino Acid Sequences

A novel method is presented for the investigation of protein properties of sequences using Ramanujan Fourier Transform (RFT). The new methodology involves the preprocessing of protein sequence data by numerically encoding it and then applying the RFT. The RFT is based on projecting the obtained numerical series on a set of basis functions constituted by Ramanujan sums (RS). In RS components, periodicities of finite integer length, rather than frequency, (as in classical harmonic analysis) are considered. The potential of the new approach is documented by a few examples in the analysis of hydrophobic profiles of proteins in two classes including abundance of alpha-helices (group A) or beta-strands (group B). Different patterns are provided as evidence. RFT can be used to characterize the structural properties of proteins and integrate complementary information provided by other signal processing transforms.

Odd Ramanujan Sums of Complex Roots of Unity

A special class of odd-symmetric length-4N periodic signals is studied, and it is shown how the odd Ramanujan sums are used as weighting coefficients to compute their pure imaginary discrete Fourier transform (DFT) integer-valued coefficients. The odd Ramanujan sum, being the sums of complex roots of unity, can be calculated either using closed-form formulas or computed recursively through the impulse response of a derived infinite impulse response (IIR) filter. This special class of odd-symmetric signals and odd Ramanujan sums can be combined together with the previous even-symmetric special class signals and the well-known even Ramanujan sums as a useful tool for signal processing

Carmichael number with three prime factors.

Let $C_3(x)$ be the number of Carmichael numbers $n\le x$ having exactly 3 prime factors. It has been conjectured that $C_3(x)$ is of order $x^{1/3}(\log x)^{-1/3}$ exactly. We prove an upper bound of order $x^{7/20+\varepsilon}$, improving the previous best result due to Balasubramanian and Nagaraj, in which the exponent $7/20$ was replaced by $5/14$. The proof combines various elementary estimates with an argument using Kloosterman fractions, which ultimately relies on a bound for the Ramanujan sum.

Ramanujan sums via generalized Möbius functions and applications

A generalized Ramanujan sum (GRS) is defined by replacing the usual Möbius function in the classical Ramanujan sum with the Souriau‐Hsu‐Möbius function. After collecting basic properties of a GRS, mostly containing existing ones, seven aspects of a GRS are studied. The first shows that the unique representation of even functions with respect to GRSs is possible. The second is a derivation of the mean value of a GRS. The third establishes analogues of the remarkable Ramanujan′s formulae connecting divisor functions with Ramanujan sums. The fourth gives a formula for the inverse of a GRS. The fifth is an analysis showing when a reciprocity law exists. The sixth treats the problem of dependence. Finally, some characterizations of completely multiplicative function using GRSs are obtained and a connection of a GRS with the number of solutions of certain congruences is indicated.

On a general form of meet matrices associated with incidence functions

Let P=(P,≤, ∧) be a meet-semilattice with least element 0 such that every principal order ideal is finite. We define the function Ψ on P × P by where f , g and h are incidence functions of P. We calculate the determinant and the inverse of the matrix [Ψ (x i ,x j )], where S={x 1,x 2, …,x n } is a meet-closed or lower-closed subset of P and h∈ L S . Here LS is the class of incidence functions defined by We apply the results to meet matrices and obtain known determinant formulae and new inverse formulae for them. These results also concern GCD matrices, which are number-theoretic special cases of meet matrices. We also apply the results to the matrix [C(x i ,x j )], where C(m, n) is the usual Ramanujan's sum.

Circular summation of theta functions in Ramanujan's Lost Notebook

In this paper, we prove Ramanujan's circular summation formulas previously studied by S.S. Rangachari, S.H. Son, K. Ono, S. Ahlgren and K.S. Chua using properties of elliptic and theta functions. We also derive identities similar to Ramanujan's summation formula and connect these identities to Jacobi's and Dixon's elliptic functions. At the end of the paper, we discuss the connection of our results with the recent thesis of E. Conrad.

Ramanujan sums and discrete Fourier transforms

A special class of even-symmetric periodic signals is introduced. The most distinctive feature of these signals is that their real-valued Fourier coefficients can be calculated by forming a weighted average of the signal values using integer-valued coefficients. The signals arise from number-theoretic concepts concerning a class of functions called even arithmetical functions. The integer-valued weighting coefficients, being sums of complex roots of unity, are the Ramanujan sums and may be computed recursively or through closed-form arithmetical relations. The recursive method of computation is based on the cyclotomic polynomials and is described in detail. If the signal values are integers, the computation of the discrete Fourier transform (DFT) coefficients of this class of signals can be performed in an exact quantization-error-free manner by performing arithmetical operations on integers. The theoretical development is supplemented by concrete examples.

Some Identities Involving Certain Hardy Sums and Ramanujan Sum

The main purpose of this paper is, using the mean-value theorem of Dirichlet l-functions, to study the distribution properties of the hybrid mean value involving certain Hardy sums and Ramanujan sum, and give four interesting identities.

The hyperbolic, the arithmetic and the quantum phase

We develop a new approach of the quantum phase in an Hilbert space of finite dimension which is based on the relation between the physical concept of phase locking and mathematical concepts such as cyclotomy and the Ramanujan sums. As a result, phase variability looks quite similar to its classical counterpart, having peaks at dimensions equal to a power of a prime number. Squeezing of the phase noise is allowed for specific quantum states. The concept of phase entanglement for Kloosterman pairs of phase-locked states is introduced.

Cyclotomy and Ramanujan sums in quantum phase locking

Phase-locking governs the phase noise in classical clocks through effects described in precise mathematical terms. We seek here a quantum counterpart of these effects by working in a finite Hilbert space. We use a coprimality condition to define phase-locked quantum states and the corresponding Pegg–Barnett type phase operator. Cyclotomic symmetries in matrix elements are revealed and related to Ramanujan sums in the theory of prime numbers. The employed mathematical procedures also emphasize the isomorphism between algebraic number theory and the theory of quantum entanglement.

Ramanujan sums for signal processing of low-frequency noise.

An aperiodic (low-frequency) spectrum may originate from the error term in the mean value of an arithmetical function such as Möbius function or Mangoldt function, which are coding sequences for prime numbers. In the discrete Fourier transform the analyzing wave is periodic and not well suited to represent the low-frequency regime. In place we introduce a different signal processing tool based on the Ramanujan sums c(q)(n), well adapted to the analysis of arithmetical sequences with many resonances p/q. The sums are quasiperiodic versus the time n and aperiodic versus the order q of the resonance. Different results arise from the use of this Ramanujan-Fourier transform in the context of arithmetical and experimental signals.