
\documentclass{article}


\begin{document}

\begin{rawhtml}
<h1 align=center> Redheffer's matrix
</h1>
\end{rawhtml}


The Redheffer matrix $A(n)$ is an $n\times n$  matrix of 0's and 1's defined by $A(i,j) = 1$ if $j = 1$ or if $i$ divides $j$, and
$A(i,j) = 0$ otherwise.
Redheffer proved that $A(n)$ has $ n-[n \log 2]-1$ eigenvalues equal to 1.
Also,     A has a real eigenvalue  (the spectral radius) which is approximately $\sqrt(n)$, 
     a negative eigenvalue which is approximately $-\sqrt{n}$ and  
     the remaining eigenvalues are small.
     The Riemann hypothesis is true if and only if $\det(A) = O(n^{1/2+\epsilon})$ for every $\epsilon > 0$. 

Barrett  Forcade, Rodney, and Pollington 
[
\begin{rawhtml}
<a href="http://www.ams.org/msnmain?fn=130&s1=89j:15029&pg1=MR">MR 89j:15029</a>
\end{rawhtml}]  give an easy proof of Redheffer's theorem.
They also prove that the spectral radius of $A(n)$ is
$=n^{1/2}+\tfrac 12\log n+O(1).$  See also the paper of
Roesleren [
\begin{rawhtml}
<a href="http://www.ams.org/msnmain?fn=130&s1=87i:11111&pg1=MR">MR 87i:11111</a>
\end{rawhtml}].

Vaughan [
\begin{rawhtml}
<a href="http://www.ams.org/msnmain?fn=130&s1=94b:11086&pg1=MR">MR 94b:11086</a>
\end{rawhtml}] and [
\begin{rawhtml}
<a href="http://www.ams.org/msnmain?fn=130&s1=96m:11073&pg1=MR">MR 96m:11073</a>
\end{rawhtml}] 
determines the dominant eigenvalues with an error term $O(n^{-2/3})$ and shows that the nontrivial eigenvalues are  $\ll(\log n)^{2/5}$ (unconditionally), and $\ll\log\log(2+n)$ on Riemann's hypothesis. 

It is possible that the nontrivial eivenvalues lie in the unit disc.
\begin{rawhtml}
<p> <br>
<hr>
<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
\end{rawhtml}

\end{document}

