Does anyone know of a good resource that shows how the number theoretic functions and their Dirichlet Products are related? This is for further reading but might come in useful for my exams.
I am particularly interested in: $u(n), N(n), \tau(n), \sigma(n), \psi(n), \mu(n)$ and $I(n)$.
For example, I have been looking at the Dirichlet Product of various number theoretic functions with $u(n)$ and have derived the following results:
$(u * u)(n) = \sum_{d|n}u(d)u(n/d) = \sum_{d|n}1 = \tau(n)$
$(N*u)(n)=\sum_{d|n}N(d)u(n/d)=\sum_{d|n}d = \sigma(n)$
$(\psi*u)(n) = \sum_{d|n}\psi(d)u(n/d) = \sum_{d|n}\psi(d) = N(n)$
$(\mu*u)(n) = \sum_{d|n}\mu(d)u(n/d) = \sum_{d|n}\mu(d) = I(n)$
I'd be interested to hear about more of these functions and how they are related through the Dirichlet Product.
A "good resource" here is the book Introduction to Analytic Number Theory by Tom Apostol. This will be very useful. Chapter $2$ is about Arithmetical Functions and Dirichlet Multiplication.