TITLE: Usage of Complex Numbers in the Riemann Hypothesis.
QUESTION [9 upvotes]: I don't have a very good understanding of the Riemann Hypothesis, however that being said, could someone explain to me why complex numbers are used, instead of just using real numbers? Everything I've read talks about absolute convergence, among other things, and it seems as if the real part of the complex number is all that ends up being important. Is it written in terms of complex numbers simply so that the structure of the complex numbers can be used, or is there something that led Riemann to naturally choose the function to take in complex numbers?

REPLY [5 votes]: The function $\pi(x)$ counts the numbers of primes that are less than or equal to $x$. For example, $\pi(5) = 3$ because of the primes 2,3,5. Define
$$\Pi(x) = \sum_{i=1}^\infty \frac{\pi(x^{1/i})}{i}.$$
Then we have $\pi(x) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \Pi(x^{1/n})$. Finally define
$$\Pi_0(x) = \lim_{\varepsilon \rightarrow 0} \frac{\pi(x-\varepsilon) + \pi(x + \varepsilon)}{2}.$$
This is the same function as $\Pi$ except at the points where $\Pi$ is discontinuous. At those points, it is the average of the left hand and the right hand limits.
Riemann proved the formula
$$\Pi_0(x) = \text{Li}(x) - \sum_{\rho} \text{Li}(x^\rho) - \log(2) + \int_{x}^\infty \frac{dt}{t(t^2-1)\log(t)}$$
where
 $$\text{Li}(x) = \int_{0}^x \frac{dt}{\log(t)}.$$
All terms are well understood except for the summation. The summation in the middle is over the nontrivial zeros of the Riemann zeta function. These are necessarily not real.
Bottom line: The prime numbers are determined by complex values of the Riemann zeta function.