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Regarding My Background I have covered stuff like

1.Single Variable Calculus

2.Multivariable Calculus (Multiple Integration,Vector Calculus etc) (Thomas Finney)

3.Basic Linear Algebra Course (Containing Vector spaces,Linear Transformation)

4.Ordinary Differential Equation

5.Real Analysis (Sequences And series)

I am interested In Number theory and i am big fan of Ramanujan .I have not been through rigorous proofs before . But i want to dig deep into number theory especially area where Ramanujan was working .Anyone researcher,Professor can advice me about which preparations are needed for going into number theory and which books should i need .I will be highly obliged .I am asking this for self study or you can say pursuing research at home .

Note - All stuff i have covered is with help of youtube videos and self study .Thanks

godonichia
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  • https://www.google.co.uk/webhp?sourceid=navclient&hl=en-GB&ie=UTF-8&gws_rd=ssl#q=ramanujan&hl=en-GB&tbm=bks – JMP Feb 05 '15 at 06:49

2 Answers2

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For number theory books:

  1. Intro to Number Theory by Niven-Zuckerman-Montgomery is a good book for beginners.

  2. Intro to Probabilistic Number Theory by Van Vu - Tao is a contemporary book that introduces advanced students to sumsets, and arithmetic of primes, etc..

  3. Analytic Number Theory by Apostol.

But notice that your interest is in Ramanujan Number Theory, you want to brush up complex analysis since many identities come out of complex variable formulas. I would read all three books at the same time !.

DeepSea
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  • Thanks .So complex Analysis is preparation for this course ,but i am not familiar with rigrous Real analysis ,do i have to first do that or i can work in conjunction with books you have suggested .What do you suggest ? – godonichia Feb 05 '15 at 07:00
  • Can you suggest something about them too for beginning to advanced .Thanks – godonichia Feb 05 '15 at 07:05
  • How did you guessed about India . You are right . But i want to do self study as i earlier said – godonichia Feb 05 '15 at 07:09
  • Hmmm. . This is bit different from what i asked in my previous question – godonichia Feb 05 '15 at 07:12
  • Self-study is not good. You need to attend class at university. – DeepSea Feb 05 '15 at 07:14
  • Ok Thanks for recommendations – godonichia Feb 05 '15 at 07:16
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    Just wanted to add that it is more of a misconception that to understand Ramanujan's work you need to have idea of complex analysis. In fact Ramanujan never learnt complex analysis and got amazing results with just basic algebraic manipulations and calculus (and a big big intuition). None of the proofs offered by Ramanujan make use of complex analysis. – Paramanand Singh Feb 05 '15 at 07:59
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In case you are interested in Ramanujan's work, then you need to go for Bruce C. Berndt's Ramanujan Notebooks Vol 1-5. Some content on Ramanujan is available on my blog http://paramanands.blogspot.com/p/archives.html also. You should also have a look at two books by G. H. Hardy:

"A Course of Pure Mathematics" (best book on single variable calculus)

"An Introduction to the Theory of Numbers"

Paramanand Singh
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