…that leads to where no mathematician has gone before.

In the Mathematical Buffet of the August 2012 issue of the Girls’ Angle Bulletin, there are a bunch of pictures that show the images, under various analytic complex functions, of the 2 by 2 square centered at the origin of the complex plane.

The image above is similar, but the domain has changed and the transformation applied is the famous Riemann-Zeta function, following the suggestion of Timothy Chow. We applied the Riemann-Zeta function to the rectangular domain in the complex plane whose four corners are 0.3, 0.3 + 27i, 0.7 + 27i, and 0.7. This domain was painted with blue horizontal bands and orange vertical bands. There’s a horizontal band every eighth of a unit and a vertical band every tenth of a unit. To make this image, we used MATLAB, a powerful suite of mathematical software produced by MathWorks.

The Riemann-Zeta function is the analytic continuation of the function first studied by Leonhard Euler:

$\zeta(s) = \Sigma_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\dots$.

The Riemann hypothesis is the conjecture, made by Bernhard Riemann in 1859, that this function only has nontrivial zeros along the line Re z= 1/2.  According to Enrico Bombieri, many mathematicians consider the Riemann hypothesis as “probably the most important open problem in pure mathematics today.” The orange band that runs down the middle of the strip represents this line for imaginary values between 0 and 27. In 1914, G. H. Hardy proved that there are, in fact, infinitely many zeros along this line, so if this strip extended vertically upward, the path would wind round and round, the middle orange line revisiting zero infinitely many times…it’s the ultimate long and winding road! According to Wikipedia, Paul McCartney had something else in mind for the long and winding road…possibly an actual road in Scotland. Besides, this long and winding road probably doesn’t end in someone’s heart, unless it’s the heart of the structure of prime numbers. Can you traverse its distance and verify that the only places where it crosses over zero is along its central orange band?

Zooming in on zero. The central orange band self-intersects three times in the center of the inset marking the location of zero in the complex plane.

Three zeroes fall within the domain used, and, if you look closely, you will see that the middle orange band does self-intersect 3 times where zero is. For more known zeroes, visit Andrew Odlyzko’s website.

For a wonderful introduction to the Riemann-Zeta function and the Riemann hypothesis, we recommend this lecture by Jeff Vaaler delivered at the University of Texas, Austin (and introduced by John Tate).