The von Mangoldt function

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 :

\begin{displaymath} \psi(x) := \sum_{n \leq x} \Lambda (n). \end{displaymath}

Then RH is equivalent to each of the following statements

\begin{displaymath} \psi (x) =x + O(x^{1/2 + \epsilon}), \end{displaymath}

for every $\epsilon>0$ ;

\begin{displaymath} \psi (x) =x + O(x^{1/2}\log^2x); \end{displaymath}

and

\begin{displaymath} \vert\psi (x) -x\vert \leq \frac{x^{1/2}\log^2x}{8\pi}, \quad x > 73.2. \end{displaymath}

(see L. Schoenfeld [56 #15581b]).



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