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Статті в журналах з теми "Ramanujan sums"

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Автор: Grafiati

Опубліковано: 6 вересня 2023

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1

Chan, T. H., and A. V. Kumchev. "On sums of Ramanujan sums." Acta Arithmetica 152, no. 1 (2012): 1–10. http://dx.doi.org/10.4064/aa152-1-1.

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2

Laohakosol, Vichian, Pattira Ruengsinsub, and Nittiya Pabhapote. "Ramanujan sums via generalized Möbius functions and applications." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–34. http://dx.doi.org/10.1155/ijmms/2006/60528.

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Анотація:

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.

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3

Fujisawa, Yusuke. "On sums of generalized Ramanujan sums." Indian Journal of Pure and Applied Mathematics 46, no. 1 (February 2015): 1–10. http://dx.doi.org/10.1007/s13226-015-0103-1.

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4

Tóth, László. "Sums of products of Ramanujan sums." ANNALI DELL'UNIVERSITA' DI FERRARA 58, no. 1 (December 24, 2011): 183–97. http://dx.doi.org/10.1007/s11565-011-0143-3.

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5

Fowler, Christopher F., Stephan Ramon Garcia, and Gizem Karaali. "Ramanujan sums as supercharacters." Ramanujan Journal 35, no. 2 (July 17, 2013): 205–41. http://dx.doi.org/10.1007/s11139-013-9478-y.

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6

Kiuchi, Isao. "Sums of averages of generalized Ramanujan sums." Journal of Number Theory 180 (November 2017): 310–48. http://dx.doi.org/10.1016/j.jnt.2017.03.026.

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7

KIUCHI, Isao. "On Sums of Averages of Generalized Ramanujan Sums." Tokyo Journal of Mathematics 40, no. 1 (June 2017): 255–75. http://dx.doi.org/10.3836/tjm/1502179227.

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8

Chen, Guangyi, Sridhar Krishnan, and Tien D. Bui. "Matrix-Based Ramanujan-Sums Transforms." IEEE Signal Processing Letters 20, no. 10 (October 2013): 941–44. http://dx.doi.org/10.1109/lsp.2013.2273973.

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9

BERNDT, BRUCE C., and PING XU. "An Integral Analogue of Theta Functions and Gauss Sums in Ramanujan's Lost Notebook." Mathematical Proceedings of the Cambridge Philosophical Society 147, no. 2 (September 2009): 257–65. http://dx.doi.org/10.1017/s0305004109002552.

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Анотація:

AbstractOne page in Ramanujan's lost notebook is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this paper we prove all the claims made by Ramanujan about this integral.

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10

Cooper, Shaun, and Michael Hirschhorn. "Sums of Squares and Sums of Triangular Numbers." gmj 13, no. 4 (December 2006): 675–86. http://dx.doi.org/10.1515/gmj.2006.675.

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Анотація:

Abstract Motivated by two results of Ramanujan, we give a family of 15 results and 4 related ones. Several have interesting interpretations in terms of the number of representations of an integer by a quadratic form , where λ1 + . . . + λ𝑚 = 2, 4 or 8. We also give a new and simple combinatorial proof of the modular equation of order seven.

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11

XU, Ce. "EXTENSIONS OF EULER-TYPE SUMS AND RAMANUJAN-TYPE SUMS." Kyushu Journal of Mathematics 75, no. 2 (2021): 295–322. http://dx.doi.org/10.2206/kyushujm.75.295.

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12

Grytczuk, Aleksander. "On Ramanujan sums on arithmetical semigroups." Tsukuba Journal of Mathematics 16, no. 2 (December 1992): 315–19. http://dx.doi.org/10.21099/tkbjm/1496161965.

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13

Yadav, Devendra Kumar, Gajraj Kuldeep, and S. D. Joshi. "Ramanujan Sums as Derivatives and Applications." IEEE Signal Processing Letters 25, no. 3 (March 2018): 413–16. http://dx.doi.org/10.1109/lsp.2017.2721966.

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14

CHEN, GUANGYI, SRIDHAR KRISHNAN, and TIEN D. BUI. "RAMANUJAN SUMS FOR IMAGE PATTERN ANALYSIS." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 01 (December 2013): 1450003. http://dx.doi.org/10.1142/s0219691314500039.

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Анотація:

Ramanujan Sums (RS) have been found to be very successful in signal processing recently. However, as far as we know, the RS have not been applied to image analysis. In this paper, we propose two novel algorithms for image analysis, including moment invariants and pattern recognition. Our algorithms are invariant to the translation, rotation and scaling of the 2D shapes. The RS are robust to Gaussian white noise and occlusion as well. Our algorithms compare favourably to the dual-tree complex wavelet (DTCWT) moments and the Zernike's moments in terms of correct classification rates for three well-known shape datasets.

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15

Li, Wen-Ch'ing Winnie, and Keqin Feng. "Character sums and abelian Ramanujan graphs." Journal of Number Theory 41, no. 2 (June 1992): 199–217. http://dx.doi.org/10.1016/0022-314x(92)90120-e.

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16

Samadi, S., M. O. Ahmad, and M. N. S. Swamy. "Ramanujan sums and discrete Fourier transforms." IEEE Signal Processing Letters 12, no. 4 (April 2005): 293–96. http://dx.doi.org/10.1109/lsp.2005.843775.

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17

Sugavaneswaran, Lakshmi, Shengkun Xie, Karthikeyan Umapathy, and Sridhar Krishnan. "Time-Frequency Analysis via Ramanujan Sums." IEEE Signal Processing Letters 19, no. 6 (June 2012): 352–55. http://dx.doi.org/10.1109/lsp.2012.2194142.

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18

Yamasaki, Yoshinori. "Arithmetical properties of multiple Ramanujan sums." Ramanujan Journal 21, no. 3 (February 25, 2010): 241–61. http://dx.doi.org/10.1007/s11139-010-9223-8.

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19

Alkan, Emre. "Distribution of averages of Ramanujan sums." Ramanujan Journal 29, no. 1-3 (September 20, 2012): 385–408. http://dx.doi.org/10.1007/s11139-012-9424-4.

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20

Shiue, Peter J. S., Anthony G. Shannon, Shen C. Huang, and Jorge E. Reyes. "A generalized computation procedure for Ramanujan-type identities and cubic Shevelev sum." Notes on Number Theory and Discrete Mathematics 29, no. 1 (March 2023): 98–129. http://dx.doi.org/10.7546/nntdm.2023.29.1.98-129.

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Анотація:

A generalized Computation procedure for construction of the Ramanujan-type from a given general cubic equation and a cosine Ramanujan-type identity is developed from detailed analyses of the properties of Ramanujan-type cubic equations. Examples are provided together with cubic Shevelev sums.

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21

Elumalai, Karthikeyan, Devender Kumar Yadav, Anup kumar Manpura, and R. K. Patney. "Stacking Seismic Data Based on Ramanujan Sums." IEEE Geoscience and Remote Sensing Letters 17, no. 9 (September 2020): 1633–36. http://dx.doi.org/10.1109/lgrs.2019.2951300.

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22

Alkan, Emre. "Ramanujan sums are nearly orthogonal to powers." Journal of Number Theory 140 (July 2014): 147–68. http://dx.doi.org/10.1016/j.jnt.2014.01.007.

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23

Bachman, Gennady. "On an optimality property of Ramanujan sums." Proceedings of the American Mathematical Society 125, no. 4 (1997): 1001–3. http://dx.doi.org/10.1090/s0002-9939-97-03650-2.

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24

Robles, Nicolas, and Arindam Roy. "Moments of averages of generalized Ramanujan sums." Monatshefte für Mathematik 182, no. 2 (April 28, 2016): 433–61. http://dx.doi.org/10.1007/s00605-016-0907-z.

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25

Ushiroya, Noboru. "ON AN APPLICATION OF EXTENDED RAMANUJAN SUMS." JP Journal of Algebra, Number Theory and Applications 41, no. 1 (January 9, 2019): 121–36. http://dx.doi.org/10.17654/nt041010121.

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26

ALKAN, EMRE. "RAMANUJAN SUMS AND THE BURGESS ZETA FUNCTION." International Journal of Number Theory 08, no. 08 (September 19, 2012): 2069–92. http://dx.doi.org/10.1142/s1793042112501187.

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Анотація:

The Mellin transform of a summatory function involving weighted averages of Ramanujan sums is obtained in terms of Bernoulli numbers and values of the Burgess zeta function. The possible singularity of the Burgess zeta function at s = 1 is then shown to be equivalent to the evaluation of a certain infinite series involving such weighted averages. Bounds on the size of the tail of these series are given and specific bounds are shown to be equivalent to the Riemann hypothesis.

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27

Namboothiri, K. Vishnu. "Certain weighted averages of generalized Ramanujan sums." Ramanujan Journal 44, no. 3 (October 11, 2016): 531–47. http://dx.doi.org/10.1007/s11139-016-9827-8.

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28

Jutila, M. "On exponential sums involving the Ramanujan function." Proceedings of the Indian Academy of Sciences - Section A 97, no. 1-3 (December 1987): 157–66. http://dx.doi.org/10.1007/bf02837820.

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29

Murty, V. Kumar, and Dinakar Ramakrishnan. "The Manin—Drinfeld theorem and Ramanujan sums." Proceedings of the Indian Academy of Sciences - Section A 97, no. 1-3 (December 1987): 251–62. http://dx.doi.org/10.1007/bf02837828.

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30

Schlage-Puchta, Jan-Christoph. "A determinant involving Ramanujan sums and So’s conjecture." Archiv der Mathematik 117, no. 4 (July 17, 2021): 379–84. http://dx.doi.org/10.1007/s00013-021-01643-8.

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Анотація:

AbstractWe compute the determinant of a matrix containing Ramanujan sums associated to the divisors of an integer n, and use this computation to prove a weak version of So’s conjecture on circulant graphs with integral spectrum.

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31

Sardari, Naser T., and Masoud Zargar. "Ramanujan graphs and exponential sums over function fields." Journal of Number Theory 217 (December 2020): 44–77. http://dx.doi.org/10.1016/j.jnt.2020.05.010.

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32

Planat, Michel, and Haret C. Rosu. "Cyclotomy and Ramanujan sums in quantum phase locking." Physics Letters A 315, no. 1-2 (August 2003): 1–5. http://dx.doi.org/10.1016/s0375-9601(03)00942-3.

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33

Cherednik, Ivan, and Syu Kato. "Nonsymmetric Rogers-Ramanujan sums and thick Demazure modules." Advances in Mathematics 374 (November 2020): 107335. http://dx.doi.org/10.1016/j.aim.2020.107335.

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34

LE, T. A., and J. W. SANDER. "CONVOLUTIONS OF RAMANUJAN SUMS AND INTEGRAL CIRCULANT GRAPHS." International Journal of Number Theory 08, no. 07 (August 28, 2012): 1777–88. http://dx.doi.org/10.1142/s1793042112501023.

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Анотація:

There exist several generalizations of the classical Dirichlet convolution, for instance the so-called A-convolutions analyzed by Narkiewicz. We shall connect the concept of A-convolutions satisfying a weak form of regularity and Ramanujan sums with the spectrum of integral circulant graphs. These generalized Cayley graphs, having circulant adjacency matrix and integral eigenvalues, comprise a great amount of arithmetical features. By use of our concept we obtain a multiplicative decomposition of the so-called energy of integral circulant graphs with multiplicative divisor sets. This will be fundamental for the study of open problems, in particular concerning the detection of integral circulant graphs with maximal or minimal energy.

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35

Pei, Soo-Chang, and Kuo-Wei Chang. "Odd Ramanujan Sums of Complex Roots of Unity." IEEE Signal Processing Letters 14, no. 1 (January 2007): 20–23. http://dx.doi.org/10.1109/lsp.2006.881527.

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36

Zheng, Zhiyong. "On the polynomial Ramanujan sums over finite fields." Ramanujan Journal 46, no. 3 (October 20, 2017): 863–98. http://dx.doi.org/10.1007/s11139-017-9941-2.

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37

Qureshi, Mohammad Idris, and Showkat Ahmad Dar. "Generalizations and applications of Srinivasa Ramanujan’s integral associated with infinite Fourier sine transforms in terms of Meijer’s G-function." Analysis 41, no. 3 (May 19, 2021): 145–53. http://dx.doi.org/10.1515/anly-2018-0067.

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Анотація:

Abstract In this paper, we obtain analytical solutions of an unsolved integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} of Srinivasa Ramanujan [S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 1915, 75–86] with suitable convergence conditions in terms of Meijer’s G-function of one variable, by using Mellin–Barnes type contour integral representations of the sine function, Laplace transform method and some algebraic properties of Pochhammer’s symbol. Also, we have given some generalizations of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} in the form of integrals ℧ S * ⁢ ( υ , b , c , λ , y ) {\mho_{S}^{*}(\upsilon,b,c,\lambda,y)} , Ξ S ⁢ ( υ , b , c , λ , y ) {\Xi_{S}(\upsilon,b,c,\lambda,y)} , ∇ S ⁡ ( υ , b , c , λ , y ) {\nabla_{S}(\upsilon,b,c,\lambda,y)} and ℧ S ⁢ ( υ , b , λ , y ) {\mho_{S}(\upsilon,b,\lambda,y)} with suitable convergence conditions and solved them in terms of Meijer’s G-functions. Moreover, as applications of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} , the three new infinite summation formulas associated with Meijer’s G-function are obtained.

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38

MELZER, EZER. "FERMIONIC CHARACTER SUMS AND THE CORNER TRANSFER MATRIX." International Journal of Modern Physics A 09, no. 07 (March 20, 1994): 1115–36. http://dx.doi.org/10.1142/s0217751x94000510.

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Анотація:

We present a "natural finitization" of the fermionic q-series (certain generalizations of the Rogers–Ramanujan sums) which were recently conjectured to be equal to Virasoro characters of the unitary minimal conformal field theory (CFT) ℳ (p, p + 1). Within the quasi-particle interpretation of the fermionic q-series this finitization amounts to introducing an ultraviolet cutoff, which — contrary to a lattice spacing — does not modify the linear dispersion relation. The resulting polynomials are conjectured (proven, for p = 3, 4) to be equal to corner transfer matrix (CTM) sums which arise in the computation of order parameters in regime III of the r = p + 1 RSOS model of Andrews, Baxter and Forrester. Following Schur's proof of the Rogers–Ramanujan identities, these authors have shown that the infinite lattice limit of the CTM sums gives what later became known as the Rocha–Caridi formula for the Virasoro characters. Thus we provide a proof of the fermionic q-series representation for the Virasoro characters for p = 4 (the case p = 3 is "trivial"), in addition to extending the remarkable connection between CFT and off-critical RSOS models. We also discuss finitizations of the CFT modular-invariant partition functions.

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39

ROYER, EMMANUEL. "EVALUATING CONVOLUTION SUMS OF THE DIVISOR FUNCTION BY QUASIMODULAR FORMS." International Journal of Number Theory 03, no. 02 (June 2007): 231–61. http://dx.doi.org/10.1142/s1793042107000924.

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Анотація:

We provide a systematic method to compute arithmetic sums including some previously computed by Alaca, Besge, Cheng, Glaisher, Huard, Lahiri, Lemire, Melfi, Ou, Ramanujan, Spearman and Williams. Our method is based on quasimodular forms. This extension of modular forms has been constructed by Kaneko and Zagier.

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40

Liu, Hong Yan, and Wen Peng Zhang. "Some Identities Involving Certain Hardy Sums and Ramanujan Sum." Acta Mathematica Sinica, English Series 21, no. 1 (June 21, 2004): 109–16. http://dx.doi.org/10.1007/s10114-004-0345-z.

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41

Saha, Biswajyoti. "A note on arithmetical functions with absolutely convergent Ramanujan expansions." International Journal of Number Theory 12, no. 06 (July 26, 2016): 1595–611. http://dx.doi.org/10.1142/s1793042116500974.

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Анотація:

Arithmetical functions with absolutely convergent Ramanujan expansions have been recently studied in certain contexts, by the present author, Murty and many others. In this article, we aim to weaken some of their hypotheses and derive asymptotic formulas with an explicit error term for sums of the form [Formula: see text] and [Formula: see text] where [Formula: see text] and [Formula: see text] are two arithmetical functions with absolutely convergent Ramanujan expansion.

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42

Zhou, Li Na, Jia Dong Shang, and Lan Yang. "Research on Ramanujan-FMT Modulation and the Efficient Implementation Algorithm." Applied Mechanics and Materials 380-384 (August 2013): 1693–96. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.1693.

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Анотація:

Ramanujan Sums (RS) and their Fourier transforms attract many attentions in signal processing in recent years. Thanks to their non-periodic and non-uniform spectrum, RS are widely used in low-frequency noise processing, e.g., Doppler spectrum estimation and time-frequency analysis. We proved the transforms can be perfectly reconstructed under certain circumstance and built a multi-tone system using Ramanujan Fourier transforms as modulation and demodulation. This system got a lower BER in AWGN channel compare to OFDM.

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43

Shiomi, Harutaka, Tomoyoshi Shimobaba, Takashi Kakue, and Tomoyoshi Ito. "Lossless Compression Using the Ramanujan Sums: Application to Hologram Compression." IEEE Access 8 (2020): 144453–57. http://dx.doi.org/10.1109/access.2020.3014979.

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44

Coppola, Giovanni, M. Ram Murty, and Biswajyoti Saha. "Finite Ramanujan expansions and shifted convolution sums of arithmetical functions." Journal of Number Theory 174 (May 2017): 78–92. http://dx.doi.org/10.1016/j.jnt.2016.10.011.

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45

Fleming, Patrick S., Stephan Ramon Garcia, and Gizem Karaali. "Classical Kloosterman sums: Representation theory, magic squares, and Ramanujan multigraphs." Journal of Number Theory 131, no. 4 (April 2011): 661–80. http://dx.doi.org/10.1016/j.jnt.2010.10.009.

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46

SAHA, BISWAJYOTI. "Partial sums of arithmetical functions with absolutely convergent Ramanujan expansions." Proceedings - Mathematical Sciences 126, no. 3 (July 15, 2016): 295–303. http://dx.doi.org/10.1007/s12044-016-0291-6.

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47

Nowak, Werner Georg. "The average size of Ramanujan sums over quadratic number fields." Archiv der Mathematik 99, no. 5 (November 2012): 433–42. http://dx.doi.org/10.1007/s00013-012-0442-7.

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48

Han, Di, and Wenpeng Zhang. "Some new identities involving Dedekind sums and the Ramanujan sum." Ramanujan Journal 35, no. 2 (July 29, 2014): 253–62. http://dx.doi.org/10.1007/s11139-014-9591-6.

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49

McIntosh, Richard J. "The H and K Family of Mock Theta Functions." Canadian Journal of Mathematics 64, no. 4 (August 1, 2012): 935–60. http://dx.doi.org/10.4153/cjm-2011-066-0.

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Анотація:

AbstractIn his last letter to Hardy, Ramanujan defined 17 functionsF(q), |q| < 1, which he calledmockθ-functions. He observed that asqradially approaches any root of unity ζ at whichF(q) has an exponential singularity, there is aθ-functionTζ(q) withF(q) −Tζ(q) =O(1). Since then, other functions have been found that possess this property. These functions are related to a functionH(x,q), wherexis usuallyqrore2πirfor some rational numberr. For this reason we refer toHas a “universal” mockθ-function. Modular transformations ofHgive rise to the functionsK,K1,K2. The functionsKandK1appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell–Lerch sums (also called generalized Lambert series). Some relations (mock theta “conjectures”) involving mockθ-functions of even order andHare listed.

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50

Planat, M., M. Minarovjech, and M. Saniga. "Ramanujan sums analysis of long-period sequences and 1/f noise." EPL (Europhysics Letters) 85, no. 4 (February 2009): 40005. http://dx.doi.org/10.1209/0295-5075/85/40005.

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