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

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.