Where can I learn about the Riemann Zeta function/Riemann Hyposthesis?

Question

I've been very curious about the Riemann zeta function and the Riemann hypothesis and I have been wanting to learn about it on a deeper level for a while now. While I don't intend to understand everything, I want to have a pretty good understanding of it. The main factor limiting me isI've only completed courses up to Calculus. Are there any elementary sources/books/papers that I can read to get some informatino about the zeta function and necessary topics to understand it? I don't need everything, just a general overview

Answer 1

To get a "good understanding" you will need at bare minimum an elementary knowledge of number theory and complex analysis. If you just want a casual introductory book, Prime Obsession by John Derbyshire is probably what you are looking for; but, assuming you have a basic background knowledge of the Riemann Hypothesis (which I believe is implied), you may find this paper useful, as it introduces the basic concepts concisely and simply. After that, I would recommend Riemann's Zeta Function by H. M. Edwards, but it is geared towards undergraduates so it may be too advanced for you.

There are some simple conjectures that happen to be equivalent to the Riemann Hypothesis, however, these can be very misleading as they more or less hide their complexity. A good example is Robin's inequality: given $\gamma$ as the Euler-Mascheroni constant and $\sigma(n)$ the divisor function, the Riemann hypothesis is equivalent to the following inequality for $n > 5040$:

$$ \sigma(n) < e^\gamma n \log\log{n} $$

A similar bound on the divisor function was also shown to be equivalent to the Riemann hypothesis by Jeffrey Lagarias:

$$ \sigma(n) < H_n + e^{H_n}\log{H_n} \hspace{1em}\text{for } H_n = \sum^n_{k=1}\frac{1}{k} $$

Modern approaches to the Riemann hypothesis require an advanced understanding of numerous fields in mathematics. I'm not aware of any books that cover modern developments on the RH, so for advanced information you probably have to read a bunch of papers. A proof of the RH will probably use extremely advanced techniques in multiple areas of mathematics, similar to the proof of Fermat's Last Theorem.