02/19/2023, 07:03 AM
I'm interested if others have any ideas on how to extend certain arithmetic functions to the complex plane. The particular case I am interseted in the Louiville function \( \lambda(n) = (-1)^{\Omega(n)} \). Some more information on that function can be found here.
My interest in this particular function is the following. We have the very nice identity
\[ \sum \frac{\lambda(n)q^n}{1-q^n} = \sum q^{n^2}\]
Since the Louiville function is basically the characteristic function for the perfect squares. Using a residue theorem approach, we should have that \( \sum q^{n^2} \) can be continued beyond its natural boundary as the function
\[ \sum \frac{\lambda(n)q^n}{1-q^n} + \sum \frac{\lambda( \frac{2 \pi i n}{\ln(q)})}{1-e^{ -\frac{4 \pi^2}{\ln(q)}}}\]
These extra terms require looking at the Louiville function at imaginary values, so I'm curious about any approach to this.
Second, I'd also like to see what \(\ln{\lambda(z)}\) looks like on the complex plane. In particular, I think it would just be cool to see how prime factorization functions extends to the whole complex plane (for instance, if we extend the indiciator function of the primes to the complex plane, are there are 'prime' complex numbers (i.e. numbers where f(z) = 0)?)
My interest in this particular function is the following. We have the very nice identity
\[ \sum \frac{\lambda(n)q^n}{1-q^n} = \sum q^{n^2}\]
Since the Louiville function is basically the characteristic function for the perfect squares. Using a residue theorem approach, we should have that \( \sum q^{n^2} \) can be continued beyond its natural boundary as the function
\[ \sum \frac{\lambda(n)q^n}{1-q^n} + \sum \frac{\lambda( \frac{2 \pi i n}{\ln(q)})}{1-e^{ -\frac{4 \pi^2}{\ln(q)}}}\]
These extra terms require looking at the Louiville function at imaginary values, so I'm curious about any approach to this.
Second, I'd also like to see what \(\ln{\lambda(z)}\) looks like on the complex plane. In particular, I think it would just be cool to see how prime factorization functions extends to the whole complex plane (for instance, if we extend the indiciator function of the primes to the complex plane, are there are 'prime' complex numbers (i.e. numbers where f(z) = 0)?)