Almost-even functions as solutions of a linear functional equation (Q2777512)
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scientific article; zbMATH DE number 1717377
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Almost-even functions as solutions of a linear functional equation |
scientific article; zbMATH DE number 1717377 |
Statements
7 March 2002
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almost even arithmetical function
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Parseval's equation
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Ramanujan expansion
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linear functional equation
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Almost-even functions as solutions of a linear functional equation (English)
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In this short note, the author shows the following remarkable result: If for all \(n\) outside some exceptional set with upper density \(0\) the function \(n\mapsto g(n) = nf(n)\) (where \(f(n)\) is an almost even function represented by its Ramanujan expansion) satisfies the functional equation \(g(n)=g(l) + g(n-l)\) for all \(l\), \(1\leq l \leq n\), then \(g(n)=\gamma n\) identically.
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