Riemann hypothesis

Statement

All the nontrivial zeros of the Riemann zeta-function have real part $1/2$ .

For a positive real number $x$ , it states that:

$\pi (x)-\operatorname {li} (x)=O({\sqrt {x}}\log x)$ .

Here, $\pi (x)$ denotes the prime-counting function, i.e., the number of primes less than or equal to $x$ , while $\operatorname {li} (x)$ denotes the logarithmic integral function:

$\int _{0}^{x}{\frac {dx}{\log x}}$ .

In fact, more specifically, the following is an equivalent formulation of the Riemann hypothesis:

$\left|\pi (x)-\operatorname {li} (x)\right|\leq {\frac {1}{8\pi }}{\sqrt {x}}\log x,\qquad x\geq 2657$ .

Riemann hypothesis for finite fields: This is an analogue of the Riemann hypothesis for finite fields, that has been proved.