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Ramanujan Sums as Derivatives
[article]
2016
arXiv
pre-print
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Generalised multiplicative property and new method of computing Ramanujan sums are also derived in terms of interpolation. ...
Ramanujan defined a family of trigonometric sum now known as Ramanujan sums. In the last few years, Ramanujan sums have inspired the signal processing community. ...
Ramanujan Operator A new class of operators, termed here as Ramanujan operators, are defined using Ramanujan sums and it is also proved that these operators satisfies all the properties of a first derivative ...
arXiv:1611.04462v1
fatcat:vlb7xndjcfe5jaknox3bjfz6km
Derived Ramanujan Primes : R'_n
[article]
2012
arXiv
pre-print
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As an application of the existence of derived Ramanujan primes, we prove analogues for Ramanujan primes of Richert's Theorem and Greenfield's Theorem for primes. ...
For n greater than or equal to 1 we define the nth derived Ramanujan prime as the smallest positive integer with the property that if x is greater than or equal to the nth derived Ramanujan prime, then ...
Jonathan Sondow for his precious comments, reviewing drafts of this paper and most necessary corrections. ...
arXiv:1210.6991v2
fatcat:bf6fmrif3jet7e3fn5j6rusn3y
On an identity of Delange and its application to Cohen-Ramanujan expansions
[article]
2023
arXiv
pre-print
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We derive a bound as well as exact value for ∑_q|k|c_q^(s)(n)| and provide a sufficient condition for the Cohen-Ramanujan expansions to exist. ...
Delange derived the bound ∑_q|k|c_q(n)|≤ n 2^ω(k) and gave a sufficient condition for such expansions to exist. A. ...
Namboothiri derived some conditions for the existence of Ramanujan series like expansions for certain arithmetical functions in terms of the generalized Ramanujan sum (3) (henceforth called as the Cohen-Ramanujan ...
arXiv:2312.05936v1
fatcat:72in433pvzffdelzq7gmiqixsy
Ramanujan series for arithmetical functions
2013
Hardy-Ramanujan Journal
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International audience We give a short survey of old and new results in the theory of Ramanujan expansions for arithmetical functions. ...
Gadiyar and R. Padma for their comments on an earlier version of this article. ...
Introduction In 1918, Ramanujan [17] published a seminal paper entitled "On certain trigonometric sums and their applications in the theory of numbers" in which he introduced sums (now called Ramanujan ...
doi:10.46298/hrj.2013.180
fatcat:q3e7pwyt2jbufowifzk4xrpcaa
Revisiting the Formula for the Ramanujan Constant of a Series
2022
Mathematics
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We also highlight the only choice for the parameter a in the formula proposed by Hardy for a series of positive terms, so the value obtained as the Ramanujan constant agrees with other summation methods ...
Additionally, we derive the closed-formula for the Ramanujan constant of a series with the parameter chosen, under a natural interpretation of the integral term in the Euler–Maclaurin summation formula ...
Acknowledgments: The authors thank the anonymous reviewers for their careful reading and suggestions that contributed to the improvement of the manuscript. ...
doi:10.3390/math10091539
fatcat:dymkxsftavacxjb53fs5rjuoxa
An image compression method based on Ramanujan Sums and measures of central dispersion
2021
Ratio Mathematica
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This paper introduces a simple lossy image compression method based on Ramanujan Sums cq(n) and the statistical measures of numerical data such as mean and standard deviation. ...
The Ramanujan Sum cq(n) has been used in digital signal processing for a variety of applications nowadays. ...
Review of Ramanujan Sums The Ramanujan Sum c q (n) has been used by mathematicians to derive many important infinite series expansions for arithmetic functions in number theory [Apostol, 1976] . ...
doi:10.23755/rm.v41i0.683
doaj:a8ef035ded3e4516af13c116f4d62bc6
fatcat:viyly2pmczdqxck6omstx75owy
Certain weighted averages of generalized Ramanujan sums
2016
The Ramanujan journal
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2016 pre-print arXiv:1601.06879v1 fatcat:nuugttvbfrdejen7al7yucjwy4
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. ...
Being interested in the applications of the Ramanujan sums, many mathematicians later tried to generalize it to find more and more applications. One of the most popular generalization was given by E. ...
doi:10.1007/s11139-016-9827-8
fatcat:dld7bar3vzcehlm5worchf3twq
Legendre Ramanujan Sums Transform
2015
Zenodo
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CONCLUSION We propose Legendre Ramanujan sums transform (LRST) derived by applying DFT to complete generalized Legendre sequence (CGLS) matrix. ...
INTRODUCTION The Ramanujan Sums (RS) are useful to many research areas such as digital signal processing, image processing, time-frequency analysis, and so on. ...
doi:10.5281/zenodo.35798
fatcat:lfbnf3zaxja3rnf74gj4zjdj5i
An extension of the Chudnovsky algorithm
[article]
2024
arXiv
pre-print
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An evaluation due to Berndt and Chan for the modular j-invariant associated with their order-3315 extension of the Chudnovskys' Ramanujan-type series provides a key to our applications of recursions for ...
the elliptic lambda and elliptic alpha functions. ...
For the the sake of brevity, we proceed to cover required preliminaries, as below, derived from the Pi and the AGM text [10] and related to Berndt and Chan's record-setting series for 1 π [5, 7] . ...
arXiv:2403.07291v1
fatcat:4t7eeyjye5cuhdk7yr7rhtajiq
Bazı Hardy Toplamları ve Ramanujan Toplamının Ortalama Değeri Hakkında Yeni Bağıntılar
2020
Süleyman Demirel Üniversitesi Fen-Edebiyat Fakültesi Fen Dergisi
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Ethics Committee Approval and Informed Consent As the author of this study, I declare that I do not have any ethics committee approval and/or informed consent statement. ...
Conflict of Interest As the author of this study, I declare that I do not have any conflict of interest statement. ...
Regarding the properties of Hardy sums and other celebrated sums such as Ramanujan sum and Kloosterman sum, some authors [7, 8, 11, [17] [18] [19] focused on them and obtained meaningful and interesting ...
doi:10.29233/sdufeffd.702928
fatcat:w7wslhzzlvatznn2qb57cgm6tm
The $\bal $\ and $\bcl $\ Bailey Transform and Lemma
1992
Bulletin of the American Mathematical Society
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The classical case, corresponding to A¡ or equivalently U(2), contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur ...
We announce a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative "Bailey chain" concept in the setting of basic hypergeometric series very well-poised on unitary Ae ...
Let A = {A(yG)} and B = {5(j,; G)} be sequences. Let M and Af* be as above. ...
doi:10.1090/s0273-0979-1992-00268-9
fatcat:j7ga3pw3zbdfbi4bnzjm5bionm
Orthogonal Ramanujan Sums, its properties and Applications in Multiresolution Analysis
[article]
2017
arXiv
pre-print
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In this paper we have proposed Orthog- onal Ramanujan Sums (ORS) based on Ramanujan sums. In this paper we present two novel application of ORS. ...
Firstly a new representation of a finite length signal is given using ORS which is defined as Orthogonal Ramanujan Periodic Transform.Secondly ORS has been applied to multiresolution analysis and it is ...
Orthogonal Ramanujan Sums (ORS) In this section we propose a family of sequences, termed here as Orthogonal Ramanujan Sums (ORS), and discuss some of its properties. ...
arXiv:1707.06283v1
fatcat:fhi6tnm4grczdn4vc3ogfngzda
Finite Ramanujan expansions and shifted convolution sums of arithmetical functions
2017
Journal of Number Theory
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2016 pre-print arXiv:1612.03136v1 fatcat:7peawszaoncv5dbefkbc4yolny
Here we introduce the concept of finite Ramanujan expansion of an arithmetical function and extend our earlier works in this setup. ...
For two arithmetical functions f and g, we study the convolution sum of the form ∑_n < N f(n) g(n+h) in the context of its asymptotic formula with explicit error terms. ...
We would also like to thank the organisers of conferences 'Leuca 2016' and 'CNTA 2016' for their kind hospitality, where the final part of this work was done. ...
doi:10.1016/j.jnt.2016.10.011
fatcat:j7zgcdutabfx5dtmw6hpdg5uge
Page 3043 of Mathematical Reviews Vol. , Issue 84h
[page]
1984
Mathematical Reviews
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N, where D(N) denotes the number of ways in which N can be expressed as the sum of two primes and C(N)=I1,..(1—- 1/(p— 1) w5>2( P— DA p—2). The proofs are much simpler than the original ones. ...
Such a process with general k in (2) yields known multisum generalizations of the Rogers-Ramanujan identities {Andrews, The theory and applications of special functions (Madison, Wis., 1975), 191-224, ...
A reciprocity theorem for the Cohen-Ramanujan sums and its application to Cohen-Ramanujan expansions in the second variable
[article]
2023
arXiv
pre-print
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Here we prove a reciprocity result on Cohen-Ramanujan sums c_k^s(n) :=∑_h=1 (h,k^s)_s=1^k^se^2π i n h/k^s to change the position of k and n in a twisted function and use it to prove that for certain arithmetical ...
sum. ...
Namboothiri derived some conditions for the existence of Ramanujan series like expansions for certain arithmetical functions in terms of the generalized Ramanujan sum (1) (henceforth called as the Cohen-Ramanujan ...
arXiv:2312.05938v1
fatcat:nmlewgx5lzb4bhjr3ejxptcjii
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