Ramanujan's sum - Open Access .click

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Journal of Number Theory
Letting \(c_q(n)\) to be the Ramanujan sum, in the paper under review, the authors provide higher convolutions of Ramanujan sums by computing the following limit \[ \lim_{x\to x}\frac{1}{x}\sum_{n\leq x}c_{q_1}(n+a_1)\cdots c_{q_k}(n+a_k). \] The result of the above limit is a multivariable multiplicative function, say \(f(q_1,\dots, q_k)\), for which ...
Goel, Shivani, Murty, M. Ram
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ON AN APPLICATION OF EXTENDED RAMANUJAN SUMS

JP Journal of Algebra, Number Theory and Applications, 2019
Summary: We study an application of the extended Ramanujan sums defined by \[ c_q(m,n)= \mathop{{\sum}'}\exp\,2\pi i (am+bn)/q, \] where the summation is taken over \(a\), \(b\) satisfying \(1\leqq a\), \(b\leqq q\) and \(\gcd(q, a, b)=1\). We extend \textit{W. Schramm}'s result [Integers 8, No. 1, Article A50, 7 p.
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Note on Ramanujan sums

2002
Summary: Let \[ S=\sum_{1\leq a\leq q}\Big|\sum_{_{\substack{ 1\leq n\leq q\\ (n,q)=1}}} b_n\exp(2\pi i\frac{an}q)\Big|^r, \] where \(r\geq 1\) is a real number and \((b_n)\) is a sequence of complex numbers. The author obtains a lower and upper bound for \(S\) and, moreover, gives an application of the Ramanujan sum to produce some identities.
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Ramanujan Sums and Almost Periodic Functions