
\documentclass{article}


\begin{document}

\begin{rawhtml}
<h1 align=center> The M\"obius function
</h1>
\end{rawhtml}

 

The M\"obius function $\mu (n)$ is defined as $(-1)^r$ if $n$ is a product of $r$ distinct primes, and as $0$ if the square of a prime divides $n$. Define :

$$
M(x) := \sum_{n \leq x} \mu (n).
$$

Then RH is equivalent to each of the following statements

$$
M (x) \ll x^{1/2 + \epsilon},
$$
for every positive $\epsilon$ ;

$$
M (x) \ll x^{1/2} \exp (A \log x/\log\log x);
$$
for some positive A.

Both results are due to Littlewood.\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}

