I have a decent high school knowledge of Elementary Number Theory and it is also a subject I love to study. I have a good background in Real Analysis (not Complex Analysis) and Abstract Algbera. I have a strong foundation in Probability Theory and some knowledge of Measure Theory.
I would like to know an interesting introductory book for Probabilistic Number Theory (a subject completely new to me but looks tremendously appealing). Do I need knowledge about Analytic Number Theory and Algebraic Number Theory? If so, please mention which book to start with and then how to make transitions.
Also, what are the prerequisites for Algebraic, Analytic or Probabilistic Number Theory? I am a second year undergraduate in mathematics and statistics.
You do not need any Algebraic Number Theory to start; of course there are some problems where you would need some but at the start you rather will not encounter them.
Some Analytic Number Theory seems at least very useful. As the type of questions is quite related.
A book I would recommend is "Introduction to Analytic and Probabilistic Number Theory: Third Edition" by Gérald Tenenbaum. (Note that this most recent edition is published by the AMS; it is not only more expansive than the earlier one from CUP, but is also cheaper.)
Choosing this book you could pick up the Analytic Number Theory in parallel.
The book is separated in three chapters:
So, there is a part of that book that uses complex analysis, but you can either skip the middle part or just browse this part, or learn this in parallel (you do not need much prior knowledge of Complex Analysis).
Given your background I think it will not be an easy read, but doable and rewarding.