An algebraic generalization of the Ramanujan sum An algebraic generalization of the Ramanujan sum

Ramanujan sums have attracted significant attention in both mathematical and engineering disciplines due to their diverse applications. In this paper, we introduce an algebraic generalization of Ramanujan sums, derived through polynomial remaindering. This generalization is motivated by its applicat...

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Bibliographic Details
Published in:The Ramanujan journal Vol. 68; no. 2; p. 37
Main Author: Kiran, N. Uday
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2025
Springer Nature B.V
Subjects:
Coding theory
Combinatorics
Data storage
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Polynomials
Synchronism
11P81
Restricted
Levenshtein codes
1L03
Shifted Varshamov–Tenengolts codes
11H71
Ramanujan sums
Finite trigonometric sums
products
ISSN:1382-4090, 1572-9303
Online Access:Get full text
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Summary:Ramanujan sums have attracted significant attention in both mathematical and engineering disciplines due to their diverse applications. In this paper, we introduce an algebraic generalization of Ramanujan sums, derived through polynomial remaindering. This generalization is motivated by its applications in Restricted Partition Theory and Coding Theory. Our investigation focuses on the properties of these sums and expresses them as finite trigonometric sums subject to a coprime condition. Interestingly, these finite trigonometric sums with a coprime condition, which arise naturally in our context, were recently introduced as an analogue of Ramanujan sums by Berndt, Kim, and Zahaescu. Furthermore, we provide an explicit formula for the size of Levenshtein codes with an additional parity condition (also known as Shifted Varshamov-Tenengolts deletion correction codes), which have found many interesting applications in studying the Little-Offord problem, DNA-based data storage and distributed synchronization. Specifically, we present an explicit formula for a particularly important open case SVT t , b ( s ± δ , 2 s + 1 ) for s or s + 1 which are divisible by 4 and for small values of δ .
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ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-025-01192-6