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.



Arithmetical function comparable to sine function
The [Liouville function](https://en.wikipedia.org/wiki/Arithmetic_function#%CE%BB(n)_%E2%80%93_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 $[-1,1]$.


![](https://ilorentz.org/beenakker/MO/LiouvilleLambda.png)
