On near orthogonality of certain \(k\)-vectors involving generalized Ramanujan sums

The near orthgonality of certain \(k\)-vectors involving the Ramanujan sums were studied by E. Alkan in [J. Number Theory, 140:147--168 (2014)]. Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by E. Cohen in [Duke Math. J., 16(2):85--90 (1949)]...

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Bibliographic Details
Published in:arXiv.org 2023-12
Main Authors: Thomas, Neha Elizabeth, Namboothiri, K Vishnu
Format: Article
Language:English
Subjects:
Lower bounds
Number theory
Orthogonality
Online Access:Get full text
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Summary:The near orthgonality of certain \(k\)-vectors involving the Ramanujan sums were studied by E. Alkan in [J. Number Theory, 140:147--168 (2014)]. Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by E. Cohen in [Duke Math. J., 16(2):85--90 (1949)]. We also prove that the weighted average \(\frac{1}{k^{s(r+1)}}\sum \limits_{j=1}^{k^s}j^rc_k^{(s)}(j)\) remains positve for all \(r\geq 1\). Further, we give a lower bound for \(\max\limits_{N}\left|\sum \limits_{j=1}^{N^s}c_k^{(s)}(j) \right|\).
ISSN:2331-8422