Dirichlet Series and the Riemann Zeta Function

When $s$ is a positive real number greater than $1$, the Riemann zeta function $\zeta$ is defined by the series $\zeta(s) = 1^{-s} + 2^{-s} + 3^{-s} + \dots$.

For example, $\zeta(1) = \infty$ since the harmonic series diverges. On the other hand, by the $p$-series test, $\zeta(s)$ evaluates to a finite positive number whenever $s > 1$.

Also, there are many surprisingly simple values of $\zeta$. For example, $\zeta(2) = \pi^2/6$ (known as the Basel problem, giving the sum of the reciprocals of the perfect squares) and $\zeta(4) = \pi^4/90$ (which actually appears in the derivation of the Stefan-Boltzmann law in physics).

Actually, the Riemann zeta function can be defined in a meaningful way for any complex number $s\neq 1$. The famous Riemann hypothesis is the deep conjecture that all zeros of $\zeta$ (complex numbers $s$ such that $\zeta(s) = 0$) have real part exactly $1/2$.

All of this is connected to number theory: for instance, the celebrated prime number theorem is equivalent to the fact that all zeros of $\zeta$ have real part not equal to $1$. Here's a question to help us see why there might be a connection to prime numbers.