Saturday, July 19, 2014

[math.NT] [hep-th] [math-ph] A theory for the zeros of Riemann Zeta and other L-functions. Part - 1 of 3

http://arxiv.org/abs/1407.4358

Overview: This paper gives a whopping 97 page overview of the Riemann Zeta function, its impact in physics, and how physics could potentially influence the Riemann-$\zeta$ hypothesis! (If you don't know about this, its importance, or history, you should. It is one of the millennium prize problems. See http://en.wikipedia.org/wiki/Millennium_Prize_Problems) I will certainly not be reviewing the entire work, since some portions delve into number theoretic territory and it is not my intention to cover this. When I first skimmed this paper though, my excitement grew uncontrollably because it gives some wonderful exposition on how this very special function plays a significant role in physics. I thought it would be a great piece to review as my first post since the material can be presented at an approachable level for advanced undergraduates and it has some really deep ideas. Just for a brief taste of what I am talking about, recall that the Riemann Zeta function is defined as 
$$ \zeta (s) := \sum_{n \in \mathbb{N}}\frac{1}{n^s}, \;\;\;\;\; s \in \mathbb{C} $$
Now because the $\zeta$-function is a function of a complex parameter, we can use techniques of complex analysis to study it and in particular we can use something called analytic continuation.  I will explain this concept later when it comes up. But for now, just understand that it almost like a magic wand. Note that $\zeta (-1)$ is divergent (or that's what we think). Well because we can use the magic of analytic continuation, we may extend the $\zeta$-function to another portion of the complex plane where $\zeta (-1)$ gives a finite values. The answer will be $- 1/12$. Look familiar? It should! Because Euler showed that 
$$\sum n = 1 + 2 + 3 + \cdots = -1/12$$
using some fancy tricks with partial sums and we've all (hopefully) heard this cute story going through our educational careers. But the $\zeta$-function gives us a much deeper understand of why this identity is true (please note that this is not the only way of deriving the identity, ask me if you want to know other ways).

But okay...this is nice any cute but why do we physicists care? Because divergences show up ALL the time in physics and we now have a meaningful way of making these sums finite! The general procedure of what I just did is known as $\zeta$-regularization. And guess what, the results we get are physical! (needs citations)

Anyway, I just wanted to get everyone reading this interested. Now lets get to the paper and start looking at what it has to offer in the physical realm... which will be in the next post. Coming soon. 

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