Book reference for studying Dirichlet Convolution

Question

Now I am studying elementary number theory, I am interested in arithmetic function, I have studied Burton's Number Theory but I can't find Dirichlet Convolution as a particular topic, I will be highly glad if anyone suggest some good book to study arithmetic function where I can find Dirichlet Convolution in details with some theorem like 'the set of all arithmetic function forms integral domain under pointwise addition and Dirichlet Convolution'. Thank you.

Answer 1

You can look for it in Introduction to Analytic Number Theory by Tom Apostol[Chapter 2], and Introduction to Analytic and Probabilistic Number Theory by Gerald Tenenbaum [Chapter I.2 in 3rd Edition]. These both give a good insight to the topic.

Answer 2

Dirichlet convolution is already defined at the beginning of Introduction to Arithmetical Functions by Paul J. McCarthy. This book presents a lot of different arithmetical functions providing a nice introduction to the subject. The chapter 4 Generalizations of Dirichlet Convolution considers $K$-convolutions: $f\star_K g$, defined as \begin{align*} \left(f\star_K g\right)(n)=\sum_{d|n}K(n,d)f(d)g(n/d)\qquad\qquad \text{for all}\ n \end{align*} which might also be of interest.