Prerequisites for understanding Riemann's zeta function?

In summary, the prerequisites for learning about the theory behind Riemann's zeta function are a basic understanding of real analysis, complex analysis, and number theory.
  • #1
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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  • #2
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.
 
  • #3
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.
 
  • #4
You'll certainly need analytic number theory, so unless you're already familiar you should add that to the list.
 

Related to Prerequisites for understanding Riemann's zeta function?

1. What is Riemann's zeta function?

Riemann's zeta function is a mathematical function named after Bernhard Riemann, a German mathematician. It is defined by the infinite series: ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ... , where s is a complex variable. It is also denoted as ζ(s) = ∑n = 1∞ 1/ns.

2. What are the prerequisites for understanding Riemann's zeta function?

To understand Riemann's zeta function, one should have a strong understanding of complex analysis, including concepts such as analytic continuation and the Cauchy integral theorem. Knowledge of number theory, including prime numbers and the concept of convergence, is also necessary. Familiarity with basic calculus, including derivatives and integrals, is also helpful.

3. What is the significance of Riemann's zeta function?

Riemann's zeta function has many applications in number theory, including the study of prime numbers and the distribution of primes. It also has connections to other areas of mathematics, such as physics and cryptography. Additionally, the Riemann hypothesis, a conjecture about the zeros of the zeta function, is considered one of the most important unsolved problems in mathematics.

4. How is Riemann's zeta function related to the Riemann hypothesis?

The Riemann hypothesis states that all nontrivial zeros of the Riemann zeta function lie on the critical line s = 1/2. This conjecture has far-reaching consequences in number theory and has been extensively studied by mathematicians. Proving or disproving the Riemann hypothesis would have significant implications for our understanding of the distribution of prime numbers.

5. Are there any real-life applications of Riemann's zeta function?

Riemann's zeta function has been used in various fields such as physics, engineering, and cryptography. In physics, it is used to study the energy levels of quantum systems. In engineering, it has been used to design efficient algorithms for digital signal processing. In cryptography, the Riemann zeta function is used in the construction of public-key cryptosystems. Additionally, the Riemann hypothesis, if proven true, could have practical applications in the field of data encryption and security.

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