The Ramanujan sum and Chebotarev densities

The Ramanujan sum c q ( n ) has been used by mathematicians to derive many important infinite series expansions for arithmetic-functions in number theory. Interestingly, this sum has many properties which are attractive from the point of view of digital signal processing. One of these is that c q ( n ) is periodic with period q , and another is that it is always integer-valued in spite of the presence of complex roots of unity in the definition. Engineers and physicists have in the past used the Ramanujan-sum to extract periodicity information from signals. In recent years, this idea has been developed further by introducing the concept of Ramanujan-subspaces. Based on this, Ramanujan dictionaries and filter banks have been developed, which are very useful to identify integer-valued periods in possibly complex-valued signals. This paper gives an overview of these developments from the view point of signal processing. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

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Ramanujan-sum expansions for finite duration (FIR) sequences

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) ◽

10.1109/icassp.2014.6854540 ◽

2014 ◽

Cited By ~ 14

Author(s):

P. P. Vaidyanathan

Keyword(s):

Finite Duration ◽

Ramanujan Sum

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Explicit formulas of a generalized Ramanujan sum

International Journal of Number Theory ◽

10.1142/s1793042116500238 ◽

2016 ◽

Vol 12 (02) ◽

pp. 383-408 ◽

Cited By ~ 1

Author(s):

Patrick Kühn ◽

Nicolas Robles

Keyword(s):

Prime Number ◽

Riemann Hypothesis ◽

Prime Number Theorem ◽

Explicit Formulas ◽

Ramanujan Sum ◽

Generalized Ramanujan Sum

In this paper, explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.

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THE DIMENSIONS OF CYCLIC SYMMETRY CLASSES OF POLYNOMIALS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500850 ◽

2013 ◽

Vol 13 (02) ◽

pp. 1350085 ◽

Cited By ~ 3

Author(s):

YOUSEF ZAMANI ◽

ESMAEIL BABAEI

Keyword(s):

Cyclic Subgroup ◽

Möbius Function ◽

Cyclic Symmetry ◽

Ramanujan Sum ◽

Symmetry Classes ◽

Mobius Function ◽

Special Cases ◽

Generalized Ramanujan Sum

The dimensions of the symmetry classes of polynomials with respect to a certain cyclic subgroup of Sm generated by an m-cycle are explicitly given in terms of the generalized Ramanujan sum. These dimensions can also be expressed in terms of the Euler ϕ-function and the Möbius function for some special cases.

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Phasor Estimation and Disturbance Detection in Power Systems by Ramanujan Sum

10.1109/pesgm46819.2021.9638126 ◽

2021 ◽

Author(s):

Bandi Ravi Kumar ◽

Abheejeet Mohapatra ◽

Saikat Chakrabarti

Keyword(s):

Power Systems ◽

Ramanujan Sum ◽

Phasor Estimation ◽

Disturbance Detection

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Carmichael number with three prime factors.

10.46298/hrj.2007.156 ◽

2007 ◽

Vol Volume 30 ◽

Author(s):

D R Heath-Brown

Keyword(s):

Upper Bound ◽

Carmichael Numbers ◽

Ramanujan Sum ◽

Prime Factors ◽

International Audience

International audience Let $C_3(x)$ be the number of Carmichael numbers $n\le x$ having exactly 3 prime factors. It has been conjectured that $C_3(x)$ is of order $x^{1/3}(\log x)^{-1/3}$ exactly. We prove an upper bound of order $x^{7/20+\varepsilon}$, improving the previous best result due to Balasubramanian and Nagaraj, in which the exponent $7/20$ was replaced by $5/14$. The proof combines various elementary estimates with an argument using Kloosterman fractions, which ultimately relies on a bound for the Ramanujan sum.

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