chapter 07
The zeros of zeta
Where the function becomes zero, structure begins to appear.
Where does zeta vanish?
A zero is an input $s$ satisfying $\zeta(s)=0$. The extended zeta function has easy-to-describe zeros at
These are called the trivial zeros. The mysterious zeros are complex and live in the vertical region $0<\operatorname{Re}(s)<1$, called the critical strip.
Symmetry narrows the mystery
The functional equation for zeta creates symmetry. Nontrivial zeros reflect across the real axis and across the line $\operatorname{Re}(s)=1/2$. A zero off the middle line would therefore bring reflected companions.
We know there are infinitely many zeros on the middle line. We also know there are no nontrivial zeros outside the critical strip. What we do not know is whether even one lies inside the strip but away from its centre.
Zeros make prime-counting ripple
Riemann found formulas in which the smooth estimate for prime counts is corrected by oscillating terms contributed by zeta’s zeros. Their imaginary parts control frequencies; their real parts control the size of the oscillations.