On Ramanujan Sums of a Real Variable and a New Ramanujan Expansion for the Divisor Function
| Autor: | Matthew S. Fox, Chaitanya Karamchedu |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Algebra and Number Theory
Mathematics - Number Theory Generalization Mathematics::Number Theory 010102 general mathematics Divisor function 0102 computer and information sciences Absolute convergence 01 natural sciences Ramanujan's sum Combinatorics symbols.namesake Number theory Integer 010201 computation theory & mathematics Convergence (routing) FOS: Mathematics symbols Number Theory (math.NT) 0101 mathematics Mathematics Real number |
| Popis: | We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also construct a new Ramanujan expansion for the divisor function. While our expansion is amenable to a continuous and absolutely convergent real variable generalization, it only interpolates the divisor function locally on $\mathbb{R}$. 6 pages |
| Databáze: | OpenAIRE |
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