RiemannZeros

The Riemann zeros are the complex numbers $s$ with non-zero imaginary part that satisfy $\zeta(s) = 0$ where $\zeta(s)$ is the Riemann zeta function defined by:

$$ \zeta(s) = 1^s + 2^{-s} + 3^{-s} + 4^{-s} + \cdots = \sum_{n=1}^{\infty} n^{-s} $$

This infinite series converges for $\text{Re}(s) > 1$, but the function can be analytically continued to other values of $s$, except for $s = 1$, where it has a simple pole.

The non-trivial zeros of the zeta function lie in the critical strip where $0 < \text{Re}(s) < 1$, and the Riemann Hypothesis is the conjecture that all such non-trivial zeros have real part $\text{Re}(s) = \frac{1}{2}$. If the real part of $s$ is $\frac{1}{2}$, the zeros are called "on the critical line."

In other words, the Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function are of the form:

$$ s = \frac{1}{2} + iy_n $$

where $y_n$ is a real number. The first few imaginary parts of non-trivial zeros on the critical line are:

$$y_1 = 14.13472514\dots$$

$$y_2 = 21.02203964\dots$$

$$y_3 = 25.01085758\dots$$