How are zeros of Riemann Zeta function related to distribution of prime numbers? What does 'proving all non-trivial zeros lie on the critical line' say about distribution of zeros on the critical
![Riemann's functional equation for the zeta function | Functions of a Complex Variable | Mathematics | MIT OpenCourseWare Riemann's functional equation for the zeta function | Functions of a Complex Variable | Mathematics | MIT OpenCourseWare](https://ocw.mit.edu/courses/18-112-functions-of-a-complex-variable-fall-2008/b039f5767c57072ec922e985c5e3749a_18-112f08.jpg)
Riemann's functional equation for the zeta function | Functions of a Complex Variable | Mathematics | MIT OpenCourseWare
![SOLVED:The Riemann zeta-function ζis defined by ζ(x) = ∑n = 1^∞ (1)/(n^x) and is used in number theory to study the distribution of prime numbers. What is the domain of ζ? SOLVED:The Riemann zeta-function ζis defined by ζ(x) = ∑n = 1^∞ (1)/(n^x) and is used in number theory to study the distribution of prime numbers. What is the domain of ζ?](https://cdn.numerade.com/previews/ba90844f-6fa8-42d7-811e-85c42c58a773_large.jpg)
SOLVED:The Riemann zeta-function ζis defined by ζ(x) = ∑n = 1^∞ (1)/(n^x) and is used in number theory to study the distribution of prime numbers. What is the domain of ζ?
![On logarithmic integrals of the Riemann zeta-function and an approach to the Riemann Hypothesis by a geometric mean with respect to an ergodic transformation | SpringerLink On logarithmic integrals of the Riemann zeta-function and an approach to the Riemann Hypothesis by a geometric mean with respect to an ergodic transformation | SpringerLink](https://media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs40879-015-0073-1/MediaObjects/40879_2015_73_Equ79_HTML.gif)