JOURNAL ARTICLE

A reciprocity theorem for the Cohen–Ramanujan sums and its application to Cohen–Ramanujan expansions.

  • Published In: International Journal of Number Theory, 2024, v. 20, n. 10. P. 2477 1 of 3

  • Database: Academic Search Ultimate 2 of 3

  • Authored By: Sivadasan, Vinod; Namboothiri, K. Vishnu 3 of 3

Abstract

For an arithmetical function f, its Ramanujan expansion is a series expansion in the form f (n) = ∑ k = 1 ∞ a (k) c k (n) where a (k) are complex numbers and c k (n) : = ∑ m = 1 (m , k) = 1 k e 2 π i m n k is the Ramanujan sum. Here, we prove a reciprocity result on Cohen–Ramanujan sums c k s (n) : = ∑ h = 1 (h , k s) s = 1 k s e 2 π i n h k s to change the position of k and n in a twisted function and use it to prove that for certain arithmetical functions f, Cohen–Ramanujan series expansions in the form ∑ k = 1 ∞ a (k) c k (s) (n) exist if and only if expansions in the form ∑ k = 1 ∞ b (k / n) c n (s) (k) exist. [ABSTRACT FROM AUTHOR]

Additional Information

  • Source:International Journal of Number Theory. 2024/11, Vol. 20, Issue 10, p2477
  • Document Type:Article
  • Subject Area:History
  • Publication Date:2024
  • ISSN:1793-0421
  • DOI:10.1142/S1793042124501185
  • Accession Number:181152390
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