Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

If $s$ is a complex number for which $\Re s > 1$, the infinite series

$$\sum\limits_{n = 1}^{\infty} \frac{1}{n^s}$$

defines an analytic function in the domain $\{s : \Re s > 1\}$, and can in fact be extended to $\mathbb{C} \setminus \{1\}$; this extension is called the *Riemann zeta function*:

$$\zeta(s)=\frac{\Gamma(-s-1)}{2\pi i}\int_{+\infty}^{+\infty} \frac{(-x)^{s-1}}{e^x-1}dx$$

where the contour travels from $+\infty$ on the $x$-axis to a counter-clockwise circle around the origin, and back to $+\infty$ on the $x$-axis.

The Riemann zeta function also has an infinite product expansion in $\{\Re s > 1\}$, giving

$$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$$

This function also satisfies Riemann's functional equation

$$\zeta(s) = 2^s \pi^{s - 1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1 - s) \zeta(1 - s)$$

where $\Gamma$ is the gamma function.

The Riemann zeta function has so-called trivial zeros at the negative even integers $-2, -4, -6, \dots$, as well as many zeros on the line $\frac{1}{2} + it$. It is conjectured that all the non-trivial zeros of the Riemann zeta function lie on this line, and this is considered to be one of the most important open problems in mathematics.

Reference: Riemann zeta function.