I was wondering if there exists or can we construct (using known arithmetic functions) an arithmetical function that has the same behaviour of the function sine or comparable to it (I mean that oscilates, changes regularly signs, periodic if it is necessary...). Thanks in advance.
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1$\begingroup$ you may want to be explicit in what you call an arithmetic function; do you follow this definition: en.wikipedia.org/wiki/Arithmetic_function $\endgroup$– Carlo BeenakkerCommented Sep 17, 2020 at 19:40
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$\begingroup$ Yes Sir, I mean that definition. $\endgroup$– Khadija MbarkiCommented Sep 17, 2020 at 19:45
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3$\begingroup$ $f(n) = 1 - 2\cdot (n \text{ mod } 2) $ $\endgroup$– Vidit NandaCommented Sep 17, 2020 at 20:11
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$\begingroup$ Thanks for your example. $\endgroup$– Khadija MbarkiCommented Sep 17, 2020 at 20:17
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1 Answer
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The Liouville function ($-1$ to the power of the number of prime divisors of $n$) could be a candidate for a "sine-like" function. It's not periodic, but it does oscillate, changing sign infinitely often, and has the same range $[-1,1]$.
answered Sep 17, 2020 at 20:06
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$\begingroup$ Thanks Sir for your answer. $\endgroup$ Commented Sep 17, 2020 at 20:10