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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.
##### MSC:
 11K65 Arithmetic functions in probabilistic number theory 11A25 Arithmetic functions; related numbers; inversion formulas
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##### References:
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