Ramanujan Sums Research Articles

Ramanujan Sums of All Natural Numbers with Grandi Series

We are all know that the Ramanujan’s theory of calculating the sum of all the natural number. When we came to known that sum of all-natural number is ,that time everyone thinks about two things, one is how it possible that sum of positive number is negative & other is how it is very close to zero. Another research paper gives value of all natural number is then I started the study of this sum, then I find it is as zero. Quit interestingly but this value has more important than Ramanujan’s sum because I did not consider the Grandi series for calculating this sum. And then it is very easy to calculate the sum of negative integer & sum of all odd natural number & sum of all even natural number.

Ramanujan Summation for Number of Diagonals in a Polygon

Among several ideas existing in Summability Theory which deals with assigning finite values to infinite divergent series of real numbers, Ramanujan Summation is one among them. There is a well known compact formula for determining number of diagonals in a convex polygon with n sides. In this paper, we will prove a new result pertaining to determining Ramanujan Summation for the divergent series whose terms are positive integral powers of number of diagonals in an n sided convex polygon.

Sparse Ramanujan Sequences Transform based OFDM for PAPR reduction

Sparse Ramanujan Sequences (SRS) are periodic series derived from Ramanujan Sums (RS). RS is an exponential summation used to derive infinite series expansions for arithmetic functions, whose application pervades different fields of study from classical mathematics to digital signal processing. SRS has properties similar to RS such as periodicity and orthogonality. SRS transforms allow the sparse representation of signals, hence they have many potential applications, such as denoising and compression. The application of SRS transform to reduce the Peak to Average Power Ratio (PAPR) in Orthogonal Frequency Division Multiplexing (OFDM) is proposed in this paper. The Complementary Cumulative Distribution Function (CCDF) and Bit Error Rate (BER) are used to assess the PAPR reduction performance of the proposed system. The results show that the proposed SRS transform-based OFDM system significantly reduces PAPR and BER with reduced computational complexity in comparison with the existing systems.

A reciprocity theorem for the Cohen–Ramanujan sums and its application to Cohen–Ramanujan expansions

For an arithmetical function [Formula: see text], its Ramanujan expansion is a series expansion in the form [Formula: see text] where [Formula: see text] are complex numbers and [Formula: see text] is the Ramanujan sum. Here, we prove a reciprocity result on Cohen–Ramanujan sums [Formula: see text] to change the position of [Formula: see text] and [Formula: see text] in a twisted function and use it to prove that for certain arithmetical functions [Formula: see text], Cohen–Ramanujan series expansions in the form [Formula: see text] exist if and only if expansions in the form [Formula: see text] exist.

On near orthogonality of certain k-vectors involving generalized Ramanujan sums

On near orthogonality of certain k-vectors involving generalized Ramanujan sums

Ramanujan Summation for Pascal’s Triangle

The concept of Pascal’s triangle has fascinated not only professional mathematicians but also everyone interested in exploring science. Similarly, the idea of Ramanujan summation has made a revolution in mathematical research after it was introduced by Srinivasa Ramanujan. In this paper, we will provide the Ramanujan summation methods for numbers located in slant diagonals of Pascal’s triangle and derive a generalized formula for such summations.

Generalized Glaisher-Kinkelin constants and Ramanujan summation of series

Generalized Glaisher-Kinkelin constants and Ramanujan summation of series

Ramanujan Summation, Riemann Zeta Function-Riemann Hypothesis, Renormalization: Godel Self-Reference, Foundational Inconsistency/Undecidability/Uncertainty in Number Theory-Mathematics

Ramanujan Summation, Riemann Zeta Function-Riemann Hypothesis, Renormalization: Godel Self-Reference, Foundational Inconsistency/Undecidability/Uncertainty in Number Theory-Mathematics

On the Hardy-Littlewood prime tuples conjecture and higher convolutions of Ramanujan sums

On the Hardy-Littlewood prime tuples conjecture and higher convolutions of Ramanujan sums

Ramanujan sums and rectangular power sums

For a fixed nonnegative integer u and positive integer n, we investigate the symmetric function∑d|n(cd(nd))updnd, where pn denotes the nth power sum symmetric function, and cd(r) is a Ramanujan sum, equal to the sum of the rth powers of all the primitive dth roots of unity. We establish the Schur positivity of these functions for u=0 and u=1, showing that, in each case, the associated representation of the symmetric group Sn decomposes into a sum of Foulkes representations, that is, representations induced from the irreducibles of the cyclic subgroup generated by the long cycle. We also conjecture Schur positivity for the case u=2.

HIPEDAP: Energy-Efficient Hardware Accelerators for Hidden Periodicity Detection

Hidden periodicity detection (HPD) forms the basis of various emerging and complex applications such as detecting tandem repeats in DNA and protein, absence seizure detection in EEG signals, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">etc.</i> The various solutions to the period estimation problem were not satisfactorily accurate until Ramanujan sums (RS) were recently used to explore the idea of periodic decomposition of signals. The use of Ramanujan sums in hidden periodicity detection was streamlined to form Ramanujan Filter Bank (RFB), but even then its usage in the applications proved to be computationally expensive. This paper proposes HIPED <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AP</sub> , an efficient set of hardware accelerators for hidden periodicity detection applications using Ramanujan Filter Bank. HIPED <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AP</sub> is developed by proposing several incrementally efficient microarchitectures targeting improvements in different aspects of the design such as area, power, and performance. We first propose a basic microarchitecture for the accelerator, Arch-A, which we then optimize and improve in terms of area, frequency of execution or power, in incremental steps to obtain Arch-E. These design improvements stem from various optimization techniques such as different filter structure realizations, transposition, pipelining and implementation of a fast division circuit. Further, the inherent error resilience exhibited by HPD applications enables us to propose an approximate architecture, Arch-F, that synergistically applies multiple approximation techniques such as approximate adder, multiplier, and precision scaling on top of Arch-E, resulting in significant performance and energy benefits. Experimental results obtained after synthesizing and mapping the hardware implementation of the microarchitectures on 45 nm technology demonstrate that the optimized Arch-E design is able to achieve 4.7X, 8.2X and 1.7X improvements over Arch-A in terms of area, frequency of execution and power, respectively. On top of this, the approximate Arch-F design demonstrated additional power savings of 14.6% on average (and upto 32.2%) over Arch-E for almost no loss in application-level quality. Finally, across a suite of practical applications, HIPED <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AP</sub> exhibited a speed-up in the range of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$5.1 {\times } 10^{2}$</tex-math></inline-formula> X to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3.2 {\times } 10^{4}$</tex-math></inline-formula> X with a geometric mean of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3.7 {\times } 10^{3}$</tex-math></inline-formula> X compared to its software implementations.

On irrationality criteria for the Ramanujan summation of certain series

In this paper, we prove a criterion for the irrationality of certain constants which arise from the Ramanujan summation of a family of infinite divergent sums. As an application, we provide a sufficient criterion for the irrationality of the values of the Riemann zeta function in the interval [Formula: see text]. We further see that our discussion leads to a natural generalization of a result of Sondow on the irrationality criterion for the Euler–Mascheroni constant.

SL(2,C) Scheme Processing of Singularities in Quantum Computing and Genetics

Revealing the time structure of physical or biological objects is usually performed thanks to the tools of signal processing such as the fast Fourier transform, Ramanujan sum signal processing, and many other techniques. For space-time topological objects in physics and biology, we propose a type of algebraic processing based on schemes in which the discrimination of singularities within objects is based on the space-time-spin group SL(2,C). Such topological objects possess an homotopy structure encoded in their fundamental group, and the related SL(2,C) multivariate polynomial character variety contains a plethora of singularities somehow analogous to the frequency spectrum in time structures. Our approach is applied to a model of quantum computing based on an Akbulut cork in exotic R4, to an hyperbolic model of topological quantum computing based on magic states and to microRNAs in genetics. Such diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum.

An efficient way of identification of protein coding regions of eukaryotic genes using digital FIR filter governed by Ramanujan's Sum

An efficient way of identification of protein coding regions of eukaryotic genes using digital FIR filter governed by Ramanujan's Sum

When Ramanujan meets time-frequency analysis in complicated time series analysis

To handle time series with complicated oscillatory structure, we propose a novel time-frequency (TF) analysis tool that fuses the short time Fourier transform (STFT) and periodic transform (PT). Since many time series oscillate with time-varying frequency, amplitude and non-sinusoidal oscillatory pattern, a direct application of PT or STFT might not be suitable. However, we show that by combining them in a proper way, we obtain a powerful TF analysis tool. We first combine the Ramanujan sums and $l_1$ penalization to implement the PT. We call the algorithm Ramanujan PT (RPT). The RPT is of its own interest for other applications, like analyzing short signal composed of components with integer periods, but that is not the focus of this paper. Second, the RPT is applied to modify the STFT and generate a novel TF representation of the complicated time series that faithfully reflect the instantaneous frequency information of each oscillatory components. We coin the proposed TF analysis the Ramanujan de-shape (RDS) and vectorized RDS (vRDS). In addition to showing some preliminary analysis results on complicated biomedical signals, we provide theoretical analysis about RPT. Specifically, we show that the RPT is robust to three commonly encountered noises, including envelop fluctuation, jitter and additive noise.

When Ramanujan sums meet affine Fourier transform

Ramanujan Fourier transform is one of the efficient multiresolution analysis tools, but it only works for stationary or periodically characterized signals. Affine Fourier transform, a general form of Fourier transform, is suitable for nonstationary signal analysis but with fixed resolution. In this study, the Ramanujan sums are combined with the affine Fourier transform to provide affine multiresolution analysis tools for nonstationary signals. First, affine Ramanujan sums are proposed and their main properties are introduced, based on which the affine Ramanujan Fourier transform (ARFT) for infinite sequences is proposed. Then, the first type ARFT (ARFT-I) for finite sequences is obtained by truncating the ARFT. Meanwhile, the transform matrix of ARFT-I is not orthogonal in general. Thus, the affine Ramanujan subspaces are constructed by affine Ramanujan sums and their canonical circular time shifts, among which different subspaces are orthogonal. Subsequently, the second type ARFT (ARFT-II) for finite sequences is developed, where the transform matrix is block orthogonal. Independent variables of ARFTs are chirp periods, which provides novel and direct methods to estimate (hidden) chirp periods. Different simulations, such as radar moving target detection and velocity estimation, are designed to show superior performance of ARFTs than affine Fourier transform based algorithms.

On some formulae related to Euler sums

Using the Ramanujan summation method, we derive some unusual formulas for a class of Euler sums (including divergent Euler sums) similar to the classical relations due to Euler.

Speech emotion recognition using Ramanujan Fourier Transform

A novel technique is presented for the analysis of Speech Emotion Recognition (SER) using Ramanujan Fourier Transform (RFT). The unique method involves numerically encoding the speech emotion data before applying the RFT. The RFT's foundation is the projection of the obtained numerical series onto a collection of fundamental functions made up of Ramanujan sums (RS). In RS components, SER data base such as Berlin, eNTERFACE, RAVDESS, SAVEE, EMOVO, EmoFilm, and Urdu are considered for testing the accuracy. This research work proposes on RFT feature based speech emotion classification. The speech emotion samples was analyzed by Ramanujan Fourier Transform and the statistical feature extraction was carried out, fed to the machine learning classifiers. The multiclass SVM based speech emotion classification was found to be proficient, when compared with the KNN and Linear Discriminant Analysis classifiers. The algorithms are evaluated on seven data bases and the results reveals that, multiclass SVM out performs other classifiers in terms of accuracy. The RFT as a stand-alone feature recognizes speech emotion with an accuracy of 83.08% for Berlin, 82.67% for eNTERFACE’ 05, 81.79% for EmoFilm, 82.98% for RAVDESS, 82.99% for EMOVO, 84% for Urdu, and 83.75% for SAVEE databases using Multiclass SVM classifier. The outcome of this research work paves a way to the researchers in speech emotion analysis for real world applications.

Exponential sum of the first kind

This paper is mainly concerned with a new exponential sum which unifies Unitary Ramanujan sum and Unitary Nagell’s Totient function. A formula for this sum and Holder equality are obtained using the structure of certain classes of integers.

Regularity in Ramanujan Summation Function Results for the Triangular Number Series

The Triangular Number Series, defined as 1 + 2 + 3 + 4 + ..., when subjected to Ramanujan Summation, gives the known and somewhat controversial result of -1/12. The Ramanujan Summation Function is defined in such a way as to accept any such series and produce corresponding values for them, thus allowing for the aforementioned result to be obtained as well. In this paper, the Triangular Number Group Series Function is defined as a function that generates members of a sum, whereas the values of those members depend on a parameter g, representing the number of elements from the Triangular Number Series in each member of the new sum. This paper shows that the results of the Ramanujan Summation Function upon any such Triangular Number Group Series follow a regularity that is also dependent on the parameter g. Once such a regularity is obtained, the scope of g is extended to real and complex numbers as well.