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

\begin{displaymath} \rho=\beta+i\gamma . \end{displaymath}

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

\begin{displaymath}N(T)=\char93 \{\rho=\beta+i\gamma\ :\ 0<\gamma\le T\} . \end{displaymath}

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,

\begin{displaymath} N(T)=\frac{1}{2\pi} T\log\left(\frac{T}{2\pi e}\right) + \frac78 + S(T) + O(1/T) , \end{displaymath}

where

\begin{displaymath} S(T)=\frac{1}{\pi} \arg \zeta\left(\frac12 + it\right) , \end{displaymath}

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:

\begin{displaymath} N_0(T) =\char93 \{\rho=\frac12 +i\gamma\ :\ 0<\gamma\le T\} , \end{displaymath}

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 [ MR 90g:11120], who proved that

\begin{displaymath} N_0(T)\ge 0.40219 \, N(T) \end{displaymath}

if $T$ is sufficiently large.

And,

\begin{displaymath} N(\sigma,T)= \char93 \{\rho=\beta +i\gamma\ :\ \beta> \sigma\ and\ 0<\gamma\le T\} , \end{displaymath}

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 density hypothesis.



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