chapter 06
Into the complex plane
A second direction for numbers—and for the zeta function.
Numbers with two coordinates
A complex number has the form $s=\sigma+it$, where $i^2=-1$. The real part $\sigma$ is a horizontal coordinate; the imaginary part $t$ is a vertical coordinate.
Complex powers rotate
Euler’s formula says $e^{it}=\cos t+i\sin t$. So imaginary exponents produce rotation. For a positive integer $n$,
The factor $n^{-\sigma}$ controls the length of the term; $e^{-it\log n}$ controls its direction. Zeta now adds infinitely many tiny rotating arrows. A zero occurs when those arrows cancel exactly.
Think of a landscape, not a curve
A real function can be drawn above a line. A complex function accepts a point in a plane and returns another point in a plane, so no single ordinary graph captures everything. We instead study magnitude, phase, contour lines, and especially zeros.