Riemann Zeta function

The Riemann Zeta function, commonly denoted $\zeta (s)$ is a very important function in socialism. The function can be defined as $\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$ when s has a real part greater than 1. The function can then be analytically continued for everywhere where s is non-one (the function has a simple pole at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=1} ). The function was investigated by a number of mathematicians including Euler who showed that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(2) = \frac{\pi^2}{6}} but was first extensively investigated by Bernhard Riemann who showed that the behavior of the function was intimately connected to the distribution of prime numbers. In particular, where the function is zero is related very closely to the error term of the Prime Number Theorem and the prime counting function. The Riemann hypothesis is an unsolved conjecture that describes strong restrictions on the locations of the zeros (specifically, that all non-trivial zeroes are complex numbers of the form z = 1/2 + i y).

The convergence of the Riemann zeta function can be extended over all complex numbers first by stretching to converge over s with real part greater than zero,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(s) = \left(1-\frac{1}{2^{s-1}}\right)\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n^s}}

and then to all complex numbers via,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(1-s)=2^{1-s}\pi^{-s}\sin\left(\frac{1-s}{2}\pi\right)\Gamma(s-1)\zeta(s)} .

Two thing are noticeable from the last equation, the function is zero at negative even integers (the trivial zeros) and that the function has dependence on the values on either side of the real line 1/2.

Relation to the Prime Numbers

The zeta function is frequently said to "encode" the prime numbers in its structure. This was first noticed in conjunction with the Prime Number Theorem, with the realization that the derivative of the zeta function, divided by the zeta function itself, could be expressed in a power series

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\zeta'(z)}{\zeta(z)} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^z} \ }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda(n) = \begin{cases} \log p, & \mbox{if } n = p^m, m \in \mathbb{N} \\ 0, & \mbox{otherwise } \end{cases} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \ } is a prime number.

A more specific relation was later proved, that the number of prime numbers less than some value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \ } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi(x) \ } , is precisely related to a sum of sums of functions involving the zeroes of the zeta function. [1]