# Some advice is needed

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
I am a sophomore student at university and decided to take a major in Computer Science. I just read about Mathematics as a self-study. For the most part I am interested in special functions ( especially digamma and zeta functions) also I like integral transforms. I read a lot about mathematics but the problem is when I read about a certain subject I realize that I am missing something so I go and read about it which might take me weeks to fully understand it then come back to the original topic again. Originally at university I only have been taught basic calculus and multi-variable calculus. I decided recently to go deep in understanding the zeros of zeta function and its analyticity as I have a background in complex analysis. My question is can I continue like that or should I take a major in mathematics ? The topics I read about are ( special functions , differential equations , linear algebra , complex analysis , real analysis , Fourier series and transform , Laplace transform )
What else should I read about to fully understand the zeta function and its zeros ?

#### mathbalarka

##### Well-known member
MHB Math Helper
ZaidAlyafey said:
What else should I read about to fully understand the zeta function and its zeros ?
You have to have a background in Complex Analysis to learn about zeta. I don't know how much have you learned so I can't tell whether you need to learn more or not. After CA, you must learn a little basics on ring theory and field theory since algebraic number theory should be learned when you are going towards zeta. Analytic number theory is the last and the most important tool for zeta.

In other words, you have to learn Number Theory in order to understand zeta.

Well, you can approach from other direction like Physics, Complex Dynamics, etc but that would take quite long for you since you are familiar with analytical methods the most.

Hope this helps,
Balarka
.

#### ZaidAlyafey

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MHB Math Helper
Hey Balarka

My knowledge in CA is basically the topics discussed in any standard CA book , I can't claim I am an expert in that but I found it interesting enough for my especially the residue theorem.

I started reading a book ( Introduction to the theory of Riemann zeta function) I can say I have now difficulty until now , in the general understanding.

Number theory is a very general subject , I can't just go and read every thing about it as I don't have time for that, what I am thinking about is reading the book and if anything I don't get I go and read about it .

#### mathbalarka

##### Well-known member
MHB Math Helper
My knowledge in CA is basically the topics discussed in any standard CA book , I can't claim I am an expert in that but I found it interesting enough for my especially the residue theorem.
I don't know what do you mean by standard book, since I never learned anything in CA from a book. I have one, yes, but it is too restricted on Residues which isn't my area of working.

Here is a list of CA topics and tell me which do you know about :

Riemann Surface, Mittang-Leffler's expansion, Weierstrass Factorization, Analytic functions, Casorati–Weierstrass theorem.

About the book you are reading, Introduction to the theory of Riemann zeta function, the name doesn't ring the bell, can you specify the writers?

Oh, so you want to read as much as you can comprehend, right? So try Titchimarsh & Health-Brown first.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
I don't know what do you mean by standard book,
For example , "First Course in Complex Analysis " By Zill.

since I never learned anything in CA from a book. I have one, yes, but it is too restricted on Residues which isn't my area of working.
You mean you depend on google search and papers?

Here is a list of CA topics and tell me which do you know about :

Riemann Surface, Mittang-Leffler's expansion, Weierstrass Factorization, Analytic functions, Casorati–Weierstrass theorem.
I only know about Riemann Surface and Analytic functions.

About the book you are reading, Introduction to the theory of Riemann zeta function, the name doesn't ring the bell, can you specify the writers?
By Patterson.

Oh, so you want to read as much as you can comprehend, right? So try Titchimarsh & Health-Brown first.
I can't find this book online !.

#### mathbalarka

##### Well-known member
MHB Math Helper
I mean of course I learned the basics from a book but mine was only restricted to the basics and there was nothing after that, so I decided to read papers on more advanced subjects on CA.

You know about Riemann Surfaces? Wow, that's good! Do you know about elliptic curves?

Oh, yes, finding it in the internet is a pain. I guess you have to go with Patterson then. Can you specify the parts you don't understand? Maybe we can help then.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
You know about Riemann Surfaces? Wow, that's good! Do you know about elliptic curves?
No , just heard about it , I cannot read about everything I hear about , this will not end .

Oh, yes, finding it in the internet is a pain. I guess you have to go with Patterson then. Can you specify the parts you don't understand? Maybe we can help then.
Sure , I will post any thing I don't understand.