Questions on the Riemann hypothesis, a conjecture on the behavior of the complex zeros of the Riemann $\zeta$ function. You might want to add the tag [riemann-zeta] to your question as well.
For complex numbers $s$ for which $\Re s > 1$, the series
$$\sum_{n = 1}^{\infty} \frac{1}{n^s}$$
converges absolutely and defines an analytic function. The Riemann zeta function is then defined to be the analytic continuation of this function. This continuation has so-called trivial zeros at the negative even integers $-2, -4, -6, ...$ as well as many zeros on the line $\frac{1}{2} + it$. The Riemann hypothesis is a famous conjecture that all the non-trivial zeros of the Riemann zeta function lie on this line.
The Riemann hypothesis has extensive implications in number theory. It is known that the truth of the claim would give precise bounds on the error involved in the prime number theorem, as well as giving strong bounds on the growth of many arithmetic functions (such as the Mertens function). More consequences are listed here.
There has been partial progress towards proving the Riemann hypothesis. Hardy and Littlewood showed that there are infinitely many zeros on the critical line, and that has been improved to show that more than two-fifths of the zeros lie on this line. There is also numerical evidence that the conjecture is true.
