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<h1 align=center> Zero counting functions
</h1>
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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
$$
\rho=\beta+i\gamma .
$$
It is common to list the zeros with $\gamma>0$ in order of
increasing imaginary part as $\rho_1=\beta_1+i\gamma_1$,
$\rho_2=\beta_2+i\gamma_2$,.... Here zeros are repeated according
to their multiplicity.

We have the zero counting function 
$$N(T)=\#\{\rho=\beta+i\gamma\ :\ 0<\gamma\le T\} .
$$
In other words, $N(T)$ counts the number of zeros in the 
critical strip, up to height $T$.
By the functional equation and the argument principle,
$$
N(T)=\frac{1}{2\pi} T\log\left(\frac{T}{2\pi e}\right) 
+ \frac78 + S(T) + O(1/T) ,
$$
where
$$
S(T)=\frac{1}{\pi} \arg \zeta\left(\frac12 + it\right) ,
$$
with the argument obtained by continuous variation along the straight
lines from $2$ to $2+iT$ to $\frac12 + iT$.  Von Mangoldt
proved that $S(T)=O(\log T)$, so we have a fairly precise estimate
of the number of zeros of the zeta-function with height less than $T$.
Note that Von Mangoldt's estimate implies that a zero at height $T$
has multiplicity $O(\log T)$.  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:
$$
N_0(T) =\#\{\rho=\frac12 +i\gamma\ :\ 0<\gamma\le T\} ,
$$
which counts zeros on the critical line up to height~$T$.  The
Riemann Hypothesis is equivalent to the assertion $N(T)=N_0(T)$
for all $T$.  Selberg proved that $N_0(T)\gg N(T)$.  At present the 
best result of this kind is due to Conrey [
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<a href="http://www.ams.org/msnmain?fn=130&s1=90g:11120&pg1=MR">MR 90g:11120</a>
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proved that
$$
N_0(T)\ge 0.40219 \, N(T) 
$$
if $T$ is sufficiently large.

And,
$$
N(\sigma,T)=
\#\{\rho=\beta +i\gamma\ :\ \beta> \sigma\ and\ 0<\gamma\le T\} ,
$$
which counts the number of zeros in the 
critical strip up to height $T$, to the right of the $\sigma$-line.
Riemann Hypothesis is equivalent to the assertion $N(\frac12 ,T)=0$
for all $T$.

For more information on $N(\sigma,T)$, see the article on the
\begin{rawhtml}<a href="../25a">density hypothesis</a>\end{rawhtml}.
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<p> <br>
<hr>
<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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