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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Published in:arXiv.org 2023-12
Main Authors: Thomas, Neha Elizabeth, Namboothiri, K Vishnu
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Lower bounds
Number theory
Orthogonality
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description 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 \(\max\limits_{N}\left|\sum \limits_{j=1}^{N^s}c_k^{(s)}(j) \right|\).
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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 \(\max\limits_{N}\left|\sum \limits_{j=1}^{N^s}c_k^{(s)}(j) \right|\).</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Lower bounds ; Number theory ; Orthogonality</subject><ispartof>arXiv.org, 2023-12</ispartof><rights>2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). 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subjects Lower bounds
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Orthogonality
title On near orthogonality of certain \(k\)-vectors involving generalized Ramanujan sums
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