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:

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

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:

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

Related facts and conjectures

Stronger conjectures

• Generalized Riemann hypothesis
• Extended Riemann hypothesis

Weaker facts and conjectures

• Prime gap corollary to Riemann hypothesis
• Lindelof hypothesis

Related facts

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

External links

• Riemann hypothesis on Wikipedia
• Riemann hypothesis on Mathworld