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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.

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