Analytic Number Theory - 1

For those who have been following my blogs earlier, you would know that I am an amateur, self-made mathematician, and I like to write as I learn new concepts or explore old ones.

My old blog, which is now abandoned to start this new and more organized blog, can be accessed from here: Infinity & Beyond

Analytic number theory is taught as a part of higher mathematics in post-graduate Pure Science (Mathematics Hons.) courses and under-graduate engineering Courses in India.

While going through the internet, we find various sources for learning about this subject. Therefore, I will break this blog up into various pieces to cover this subject in significant details.

Coming to this, we will keep Tom M Apostol’s Introduction to Number Theory as the foundational text for this blog.

In this blog, we will cover the contents of Chapter 1 and Chapter 2 of Apostol.

Therefore, Elementary Number Theory, Fundamental Theorem of Arithmetic, Arithmetical Functions and Dirichlet Multiplication will be covered, amognst various other concepts.

It is again to be noted that, readers may take this is a disclaimer before we start, that I am writing this blog as a way to teach myself this subject and would therefore run this blog in a “note making/understanding” tone, rather than as a tutorial.

Fundamental Theorem of Arithmetic:

Many proofs of Analytic number theory makes use of the concepts that we learn in this section.

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The principle of Induction:

If $Q$ is set of integers such that,
(a) $1 \in Q$
(b) $n \in Q \text{ implies } n+1 \in Q$
then,
(c ) $\text{all integers }\geq1\text{ belongs to } Q$

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