# Math 22a Fall 2018

## 22a Linear Algebra and Vector Analysis

# Riemann Hypothesis

There is currently quite a buzz about an attempt to prove the Riemann hypothesis (i.e. Science Mag). The Riemann Hypothesis claims that all non-trivial roots of the function f(s) = ∑_{n=1}

^{∞}1/n

^{s}are on the line Re(s)=1/2. Below is a picture of the level curves of that function. We see the

**critical strip**Re(s) in [0,1]. The picture was done in this project. You can click on the picture to see a 3840 x 14813 resolution picture of the Zeta function level curves: (The picture was rendered in Mathematica in that 2016 project).] In the picture, we see the window [-1.2,1/2] for the real part and [0,100] for the imaginary part (split up in 5 strips). You see at the very bottom right of the first strip the number s=1. This is the pole of the zeta function as the harmonic series 1+1/2+1/3+1/4+ ... diverges. At other points like s=-1, one can however assign a nice finite value: 1+2+3+4+.... = -1/12 (this is called analytic continuation). If you do not believe that, compute it with Mathematica or look it up in Wolfram alpha. You see the roots of the function as black little circles. They all are on the line Re(s)=1/2. It is the most important open problem in mathematics to prove this. It appears now that also the attempt of Atiyah did not yet solve the problem. on this blog provides some links some links (September 28, 2018), especially to this entry, a computer scientist, which explains the Todd trick. And there is more explanation here from a physicist from September 24. It is not that strange that mathematicians are more cautious in expressing opinions. It is an important open problem. And it has not been proven that there is NOT a simple argument which proves the conjecture. Looking at the history of mathematics, it is unlikely. Important open problems which were left open for a long time often were solved only with new theories only. There are of course the examples, where a conjecture is solved with a counter example. But also this often requires new insight beyond brute force. John Baez has in the wake of the Atiyah announcements tweeted some example, where surprises only start to happen with large numbers. This is called the "strong law of small numbers": Example 1, Example 2, Example 3. Some believe that this could happen for the Zeta function. It could be that for extremely large value only (maybe even not accessible to us), the roots start to deviate from the line Re(s)=1/2.