Riemann hypothesis

Statement

In terms of zeros of the Riemann zeta-function

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

In terms of the distribution of prime numbers

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

$\pi (x)-\mathrm{l}\mathrm{i}(x)=O(\sqrt{x}\log x)$.

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

$\underset{0}{\overset{x}{\int }}\frac{dx}{\log x}$.

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

$|\pi (x)-\mathrm{l}\mathrm{i}(x)|\le \frac{1}{8\pi }\sqrt{x}\log x,x\ge 2657$.

Related facts and conjectures

Stronger conjectures

• Generalized Riemann hypothesis
• Extended Riemann hypothesis

Weaker facts and conjectures

• Prime gap corollary to Riemann hypothesis
• Lindelof hypothesis