Prerequisites for understanding Riemann's zeta function?

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Discussion Overview

The discussion focuses on the prerequisites necessary for understanding Riemann's zeta function, particularly from the perspective of someone with a background in physics. Participants explore the mathematical foundations required to comprehend related literature, specifically a book by H. M. Edwards.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant suggests that basic real analysis is essential, including rigorous proofs, continuity, sequences, series, limits, and various convergence theorems.
  • Complex analysis is also highlighted as necessary, with an emphasis on holomorphic functions, contour integration, and Cauchy's integral formula.
  • Another participant expresses uncertainty about the typical mathematical background of physics graduates, noting significant variability among individuals.
  • There is a mention of the need to work through manipulations and simplifications in Edwards' book, which may not be fully detailed.
  • One participant proposes studying analytic number theory as an additional prerequisite.
  • Participants discuss potential textbooks for real analysis and complex analysis, with specific titles mentioned, including works by Serge Lang, Charles Pugh, and Theodore Gamelin.
  • There is uncertainty regarding the necessity of studying number theory in relation to the zeta function.

Areas of Agreement / Disagreement

Participants generally agree on the importance of real and complex analysis as prerequisites, but there is no consensus on the necessity of number theory or the specific mathematical background of physics graduates.

Contextual Notes

Participants express varying levels of familiarity with the required mathematical concepts, indicating that individual preparedness may differ significantly. The discussion does not resolve the question of how much mathematics an average physics graduate possesses.

Who May Find This Useful

Individuals interested in studying Riemann's zeta function, particularly those with a background in physics or mathematics, may find this discussion relevant.

cragwolf
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I am wondering what are the prerequisites required for learning the theory behind Riemann's zeta function, starting from a base of mathematics that an average physics graduate might have. In particular, I want to be able to understand a book like this:

https://www.amazon.com/dp/0486417409/?tag=pfamazon01-20 by H. M. Edwards
 
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Hi cragwolf, I have this book and could understand it in my third year (of a mathematics degree).. I'm not sure how much mathematics the average physics graduate has? Judging from people I've met it must vary significantly..

The main prerequisites are basic real analysis (rigourous proofs, continuity, sequences, series and limits, various convergence theorems, possible Riemann-Stieljes integration [just how the notation works, not a proof that it exists for smooth functions etc.]) then complex analysis (you definitely need to know about holomorphic functions, countour integration, Cauchy's integral formula, intergration involving around simple poles).

I assume you probably have experience with some of the above. Edwards makes the (I think) deliberate point of missing out certain steps when manipulating equations - e.g. he says "upon applying Cauchy's integral formula we get..", and you will have to work out exactly how he applied it and the subsequent manipulations or simplifications.. but this makes it a more rewarding read when you can follow it.
 
olliemath said:
Hi cragwolf, I have this book and could understand it in my third year (of a mathematics degree).. I'm not sure how much mathematics the average physics graduate has? Judging from people I've met it must vary significantly..

Heheh, I purposely chose a rather vague measure because I'm not sure what I remember of my mathematical education. Thanks for the information. Looks like I'll need to study up on real analysis and complex analysis.

For real analysis I'm deciding between Undergraduate Analysis by Serge Lang or Real Mathematical Analysis by Charles Pugh. Rudin scares me and my wallet. For complex analysis I'm thinking that Complex Analysis by Theodore Gamelin might be the choice. Also, I'm not sure whether I should read up on number theory.
 
You'll certainly need analytic number theory, so unless you're already familiar you should add that to the list.
 

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