# Zero counting functions

Below we present the standard notation for the functions which count zeros of the zeta-function.

Zeros of the zeta-function in the critical strip are denoted

It is common to list the zeros with in order of increasing imaginary part as , ,.... Here zeros are repeated according to their multiplicity.

We have the zero counting function

In other words, counts the number of zeros in the critical strip, up to height . By the functional equation and the argument principle,

where

with the argument obtained by continuous variation along the straight lines from to to . Von Mangoldt proved that , so we have a fairly precise estimate of the number of zeros of the zeta-function with height less than . Note that Von Mangoldt's estimate implies that a zero at height has multiplicity . That is still the best known result on the multiplicity of zeros. It is widely believed that all of the zeros are simple.

A number of related zero counting functions have been introduced. The two most common ones are:

which counts zeros on the critical line up to height . The Riemann Hypothesis is equivalent to the assertion for all . Selberg proved that . At present the best result of this kind is due to Conrey [ MR 90g:11120], who proved that

if is sufficiently large.

And,

which counts the number of zeros in the critical strip up to height , to the right of the -line. Riemann Hypothesis is equivalent to the assertion for all .

For more information on , see the article on the density hypothesis.

Back to the main index for The Riemann Hypothesis.