Riemann zeta function

The Riemann zeta function (also known as the Euler–Riemann zeta function), notated as $\zeta(s)$, is a function used in complex analysis and number theory. It is defined as the analytic continuation of the series

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

which converges for all s such that $\mathrm{Re}(s) > 1$.

The Riemann hypothesis states that $\zeta(s) = 0$ iff s is a negative even integer or the imaginary part of s is 1/2.

Representation as an integral

The Riemann zeta function can be expressed as an improper integral. Consider the improper integral,

$\int_{0}^{\infty} \frac{t^{z-1}}{e^t - 1} \mathrm{d}t$

One may multiply through $\frac{e^{-t}}{e^{-t}}$ to obtain,

$\int_{0}^{\infty} \frac{e^{-t}t^{z-1}}{1 - e^{-t}} \mathrm{d}t$

From which point, considering,

$\frac{1}{1-e^{-t}} = \sum_{n \geq 0} (e^{-t})^n$

One may rewrite the integrand as,

$\int_{0}^{\infty} e^{-t}t^{z-1} \sum_{n \geq 0} e^{-nt} \mathrm{d}t$

Which can be written as, by changing the order of the operators

$\int_{0}^{\infty} \sum_{n \geq 1} e^{-nt}t^{z-1} \mathrm{d}t$

By using a substitution of, say, $x = nt$ $\left(\implies \mathrm{d}x = n\mathrm{d}t \implies \mathrm{d}t = \frac{1}{n}\mathrm{d}x\right)$, one can write this as

$\int_{0}^{\infty} \sum_{n \geq 1} \frac{1}{n} e^{-x} \left(\frac{x}{n}\right)^{z-1} \mathrm{d}x$

$\int_{0}^{\infty} \sum_{n \geq 1} \frac{1}{n} e^{-x} \left(\frac{x^{z-1}}{n^{z-1}}\right)\mathrm{d}x$

$\int_{0}^{\infty} \sum_{n \geq 1} \frac{1}{n^z} e^{-x} x^{z-1} \mathrm{d}x$

Notice that $\frac{1}{n^z}$ is independent of x, and $e^{-x}x^{z-1}$ is independent of n, meaning that our integral can be written as,

$\sum_{n \geq 1} \frac{1}{n^z} \int_{0}^{\infty} e^{-x}x^{z-1} \mathrm{d}x$.

Using the definitions of both the zeta and the gamma function, we finally have,

$\int_{0}^{\infty} \frac{t^{z-1}}{e^t - 1} \mathrm{d}t = \zeta(z)\Gamma(z)$

Finally,

$\zeta(z) = \frac{1}{\Gamma(z)} \int_{0}^{\infty} \frac{t^{z-1}}{e^t - 1} \mathrm{d}t$

Riemann Hypothesis

The zeta function has no zeros where s is greater than or equal to one. When s is less than or equal to zero, the function has zeros on even integers known as trivial zeros. The remaining zeros are between zero and one; this is known as the critical strip. The Riemann hypothesis is that all non-trivial zeros lie on the line $1/2 + it$ as $t$ ranges over all real numbers. Whether or not this is true is known as the Riemann Hypothesis.