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

2023/12/12 by Neha Elizabeth Thomas, Thomas, Neha Elizabeth, K Vishnu Namboothiri +1
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2312.07098

Abstract

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 \frac1k^s(r+1)∑ _j=1^ksj^rc_k^(s)(j) remains positve for all r≥ 1. Further, we give a lower bound for max_N|∑ _j=1^Nsc_k^(s)(j) |.

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