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<h1 align=center> The von Mangoldt function
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The von Mangoldt function $\Lambda (n)$ is defined 
as $\log p$ if $n$ is a power of a prime $p$, and $0$ in the other cases. Define :
$$
\psi(x) := \sum_{n \leq x} \Lambda (n).
$$

Then RH is equivalent to each of the following statements
$$
\psi (x) =x + O(x^{1/2 + \epsilon}),
$$
for every $\epsilon>0$ ;
$$
\psi (x) =x + O(x^{1/2}\log^2x);
$$
and
$$
|\psi (x) -x| \leq \frac{x^{1/2}\log^2x}{8\pi}, \quad x > 73.2.
$$
(see L. Schoenfeld [\begin{rawhtml}56 \#15581b\end{rawhtml}]). \begin{rawhtml}
<p> <br>
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<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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