Complex variable in the context of Critical point (mathematics)

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as $${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }$$ for Re(s) > 1, and its analytic continuation elsewhere.

The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

In mathematics, a Dirichlet L-series is a function of the form

where ${\displaystyle \chi }$ is a Dirichlet character and ${\displaystyle s}$ a complex variable with real part greater than ${\displaystyle 1}$. It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane; it is then called a Dirichlet L-function.