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Almost-even functions as solutions of a linear functional equation. (English) Zbl 0984.11041
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.
11K65 Arithmetic functions in probabilistic number theory
11A25 Arithmetic functions; related numbers; inversion formulas
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