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Addendum to: “Ramanujan’ sum for generalised integers”
Duke Mathematical Journal, 1966 openaire +2 more sources
Reciprocity in Ramanujan's Sum
Mathematics Magazine, 1986 Reciprocity laws have held a special place in the theory of numbers ever since Gauss, at the age of nineteen, proved the first number-theoretic reciprocity law, the law of quadratic reciprocity. Since that time, many reciprocity laws have been discovered.
Kenneth R. Johnson
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RAMANUJAN'S INTEGRALS AND GAUSS'S SUMS
The Quarterly Journal of Mathematics, 1936 G. N. Watson
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A New Generalization of Ramanujan's Sum
Journal of the London Mathematical Society, 1966 M.V. Subba Rao, V. C. Harris
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Some Formulas Involving Ramanujan Sums
Canadian Journal of Mathematics, 1962The purpose of this note is to establish an identity involving the cyclotomic polynomial and a function of the Ramanujan sums. Some consequences are then derived from this identity.For the reader desiring a background in cyclotomy, (2) is mentioned. Also, (4) is intimately connected with the following discussion and should be consulted.The cyclotomic ...
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Image Transmultiplexing using Ramanujan Sums
2020 12th International Conference on Computational Intelligence and Communication Networks (CICN), 2020Ramanujan Sums have recently aroused interest in signal processing community owing to its special properties. These sums have been used in many signal processing applications such as periodicity detection of sequences and bio molecules, filter bank design, formulation of new signal transforms, etc.
Deepa Abraham, Manju Manuel
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Some Identities Involving Certain Hardy Sums and Ramanujan Sum
Acta Mathematica Sinica, English Series, 2004In this paper, the authors consider the Dedekind sum, the Hardy sum, the Ramanujan sum and two other related sums which are defined by \[ S(d,c)=\sum^c_{j=1}\left(\left(\frac jc\right)\right)\left(\left(\frac {dj}c\right)\right),\quad S_3(d,c)=\sum^{c-1}_{j=1}(-1)^{j+1+[dj/c]},\quad R_c(d)=\sum^c_{b=1}{}'e^{2\pi idb/c}, \] \[ S_1(d,c)=\sum^c_{j=1}(-1)^{
Liu, Hongyan, Zhang, Wenpeng
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The Manin—Drinfeld theorem and Ramanujan sums
Proceedings of the Indian Academy of Sciences - Section A, 1987Let \(\Gamma\) be a discrete subgroup of \(\text{SL}_ 2(\mathbb R)\) such that the corresponding modular curve \(X=X(\Gamma)\) has finite volume. It is of interest to study the subgroup \(C(\Gamma)\) of \(J=\text{Jac}(X)\) generated by the divisors of degree \( 0\) supported on the cusps of \(X\).
Murty, V. Kumar, Ramakrishnan, Dinakar
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