Prerequisites for understanding Riemann's zeta function?

[cragwolf]

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

 

[olliemath]

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.

 

[cragwolf]

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.

 

[CRGreathouse]

You'll certainly need analytic number theory, so unless you're already familiar you should add that to the list.