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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

This is one of the unsolved problems throughout the 6 unsolved problems on the Millennium Prize Problem, which grants $1,000,000 to any person that could solve them correctly.

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, the challenge is to prove that the Riemann Hypothesis is true or false.

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