Puzzle 9: Solution

Q     The following picture appeared without explanation on a web page for John Derbyshire's Prime Obsession. What does it represent?

A     It is a plot of the Riemann zeta function on the boundary of the rectangle [0.4,0.6]+[0,14.5]i in the complex plane. Since the contour winds around the origin once (and the rectangle does not contain the point s=1, which is the unique pole of zeta(s)), the zeta function has a unique zero inside this rectangle. Since the complex zeros are known to be symmetric about the line Re(s)=1/2, this zero must have real part exactly equal 1/2, in accordance with the Riemann hypothesis.

It is known that this first “nontrivial zero” of zeta(s) occurs at s=1/2+it for t=14.13472514... The pole at s=1 accounts for the wide swath in the third quadrant, which corresponds to s of imaginary part less than 1.