2
votes
1answer
83 views

How to proceed doing number theory?

I'm an undergrad majoring in mathematics. Being in first year I'm still exploring new branches of mathematics and till now, It is analysis and Number theory that I've come to have a great interest ...
4
votes
0answers
187 views

Mathematics felt by Srinivasa Ramanujan

At the moment I am reading the book Ramanujan's Papers by B.J. Venkatachala, V. Vinay and C.S. Yogananda; when clarifying some doubt with a professor, he told me that Srinivasa Ramanujan used Galois ...
1
vote
1answer
191 views

What are the fertile areas of research in Analytic Number Theory?

My professor once told me that Analytic Number Theory was "dead," which at the time was something of a disappointment, and which I struggled to agree with. Surely any subject may appear inferrtile in ...
1
vote
1answer
126 views

Applications of prime-number theorem in algebraic number theory?

Dirichlet arithmetic progression theorem, or more generally, Chabotarev density theorem, has applications to algebraic number theory, especially in class-field theory. Since we might think of the ...
7
votes
2answers
129 views

Why is $G(k)$ “more fundamental” than the Hilbert-Waring function $g(k)$?

In the Wikipedia entry for Waring's problem, the section on $G(k)$ starts as: “From the work of Hardy and Littlewood, more fundamental than $g(k)$ turned out to be $G(k)$, which is defined...” There ...
10
votes
2answers
483 views

Supplemental number theory text to Montgomery and Vaughan

We already have a large list of the Best ever book on Number Theory, but I'm looking for a more targeted response for analytic number theory. Specifically, I'm taking a trip on which I may or may ...
34
votes
6answers
2k views

How hard is the proof of $\pi$ or e being transcendental?

I understand that $\pi$ and e are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
6
votes
2answers
298 views

A Question on RH relating to Prime Number theorem

Well, in a previous post regarding the explanation of Riemann Hypothesis Matt answered that: The prime number theorem states that the number of primes less than or equal to $x$ is approximately ...