Abstract
The near orthgonality of certain k-vectors involving the Ramanujan sums were studied by Alkan (J Number Theory 140:147–168, 2014). Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by Cohen (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 positive for all \(r\ge 1\). Further, we give a lower bound for \(\max \limits _{N}\left| \sum \limits _{j=1}^{N^s}c_k^{(s)}(j) \right| \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alkan, E.: On the mean square average of special values of L-functions. J. Number Theory 131(8), 1470–1485 (2011)
Alkan, E.: Distribution of averages of Ramanujan sums. Ramanujan J. 29(1–3), 385–408 (2012)
Alkan, E.: Averages of values of L-series. Proc. Am. Math. Soc. 141(4), 1161–1175 (2013)
Alkan, E.: Ramanujan sums are nearly orthogonal to powers. J. Number Theory 140, 147–168 (2014)
Alkan, E.: A generalization of the Hardy-Littlewood conjecture. Integers 22 (2022)
Apostol, T.: Introduction to Analytic Number Theory. Springer, New York (1976)
Beurling, A.: Analyse de la loi asymptotique de la distribution des nombres premiers généralisés i. Acta Math. 68(1), 255–291 (1937)
Carmichael, R.D.: Expansions of arithmetical functions in infinite series. Proc. Lond. Math. Soc. 2(1), 1–26 (1932)
Chaubey, S., Goel, S., Ram Murty, M.: On the Hardy–Littlewood prime tuples conjecture and higher convolutions of Ramanujan sums. Funct. Approx. Comment. Math. 69(2), 161–175 (2023)
Cohen, E.: An extension of Ramanujan’s sum. Duke Math. J. 16(2), 85–90 (1949)
Cohen, E.: An extension of Ramanujan’s sum II. Additive properties. Duke Math. J. 22(4), 543–550 (1955)
Cohen, E.: An extension of Ramanujan’s sum. III. Connections with totient functions. Duke Math. J. 23(4), 623–630 (1956)
Fainleib, A.S.: On the distribution of Beurling integers. J. Number Theory 111(2), 227–247 (2005)
Green, B., Tao, T.: The Möbius function is strongly orthogonal to nilsequences. Ann. Math. 175, 541–566 (2012)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, London (1979)
Ikeda, S., Kiuchi, I., Matsuoka, K.: Sums of products of generalized Ramanujan sums. J. Integer Seq. 19(2), 16–2 (2016)
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Springer, New York (1990)
Kiuchi, I.: Sums of averages of generalized Ramanujan sums. J. Number Theory 180, 310–348 (2017)
Ma, R., Wang, H., Zhang, Y.: Hybrid mean value of the character sums over the short interval [1, p 8) and the other famous sums. Indian J. Pure Appl. Math. (2023). https://doi.org/10.1007/s13226-023-00410-5
McCarthy, P.J.: Introduction to Arithmetical Functions. Springer, New York (2012)
Montgomery, H.L., Vaughan, R.C.: Multiplicative Number Theory I: Classical Theory, vol. 97. Cambridge University Press, Cambridge (2006)
Vishnu Namboothiri, K.: Certain weighted averages of generalized Ramanujan sums. Ramanujan J. 44(3), 531–547 (2017)
Nathanson, M.B.: Elementary Methods in Number Theory, vol. 195. Springer, New York (2008)
Ramanujan, S.: On certain trigonometrical sums and their applications in the theory of numbers. Trans. Camb. Philos. Soc 22(13), 259–276 (1918)
Singh, J.: Sums of products involving power sums of integers. J. Numbers (2014). https://doi.org/10.1155/2014/158351
Tóth, L.: Averages of Ramanujan sums: Note on two papers by E. Alkan. Ramanujan J. 35(1), 149–156 (2014)
Acknowledgements
The authors thank the anonymous reviewer for inviting our attention to a latest publication relevant to this paper, in addition to suggesting some changes to improve the overall accuracy of the paper. The first author thanks the Kerala State Council for Science, Technology and Environment, Thiruvananthapuram, Kerala, India for providing financial support for carrying out this research work.
Funding
The first author received financial support from the Kerala State Council for Science, Technology and Environment, Thiruvananthapuram, Kerala, India for carrying out this research work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Thomas, N.E., Namboothiri, K.V. On near orthogonality of certain k-vectors involving generalized Ramanujan sums. Ramanujan J (2024). https://doi.org/10.1007/s11139-024-00874-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11139-024-00874-x
Keywords
- Generalized Ramanujan sums
- Weighted power sums
- Near orthgonality
- Beurling type integers
- Jordan totient function
- Möbius function
- Bernoulli numbers
- Bernoulli polynomials