$$ \Psi_k(N)=\sum_{n=1}^{N} \mu(n)\mu(k-n)$$
where $\mu(n)$ is the Mobius function.
This function is interesting to me because for the case of $k=N$ it has the symmetric property of being odd with respect to the midpoint of the interval, at $N/2$.
I tried to plot the function for multiple values of $k$ on the same plot but could not figure out how to do this on wolfram alpha.
I'm trying to see if by plotting many of these functions I can see a pattern emerge. Maybe the set of functions stay within some region on average.
I'd like the answers to include a plot of the function for $k=1,2,...,N$ all on the same plot to see how different they are.
Why does a little change in $k$ create such a big change in the overall behavior of the function?
Thanks.