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Journal articles on the topic 'Ramanujan sums'

Author: Grafiati

Published: 6 September 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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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 fi

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3

Fujisawa, Yusuke. "On sums of generalized Ramanujan sums." Indian Journal of Pure and Applied Mathematics 46, no. 1 (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 (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 (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 (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 (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 (2009): 257–65. http://dx.doi.org/10.1017/s0305004109002552.

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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 (2006): 675–86. http://dx.doi.org/10.1515/gmj.2006.675.

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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 (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 (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 (2013): 1450003. http://dx.doi.org/10.1142/s0219691314500039.

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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 we

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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 (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 (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 (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 (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 (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 (2023): 98–129. http://dx.doi.org/10.7546/nntdm.2023.29.1.98-129.

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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 (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 (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 (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 (2012): 2069–92. http://dx.doi.org/10.1142/s1793042112501187.

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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 (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 (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 (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 (2021): 379–84. http://dx.doi.org/10.1007/s00013-021-01643-8.

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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 (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 (2012): 1777–88. http://dx.doi.org/10.1142/s1793042112501023.

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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 f

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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 (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 (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 (2021): 145–53. http://dx.doi.org/10.1515/anly-2018-0067.

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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 o

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38

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

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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 th

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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 (2007): 231–61. http://dx.doi.org/10.1142/s1793042107000924.

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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 (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 (2016): 1595–611. http://dx.doi.org/10.1142/s1793042116500974.

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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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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 (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 (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 (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 (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 (2012): 935–60. http://dx.doi.org/10.4153/cjm-2011-066-0.

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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 Rama

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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 (2009): 40005. http://dx.doi.org/10.1209/0295-5075/85/40005.

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