Neue Empirische Daten Über Die Zahlentheoretische Funktion sigma(n), von Sterneck, 1913. Part II

This post is about a formula that von Sterneck used to calculate values of \( M(n) \) for \( n \) up to 5,000,000. The formula is

\begin{equation} \sum_{d \le \sqrt{x}}\!\!\phantom{}^{\prime}\ M\left( \frac{x}{d} \right) + \sum_{d \le \sqrt{x}} \mu(d) \omega_P\left( \frac{x}{d} \right) - \omega_P(\sqrt{x})M(\sqrt{x}) =0 \end{equation}

where

• \( \mu \) the Möbius function
• \( M(x) = \sum_{ n \le x} \mu(n) \) the Mertens function
• \( \omega_P(x) = \left\{ k \in\mathbb{N} \middle| 1 \le k \le x \text{ and } p_1 \nmid k \text{ and } \cdots \text{ and } p_j \nmid k\right\} \) with \(P = \left\{ p_1, \ldots p_j \right\}\) a non-empty set of primes
• \(\sum_{k \le x}^{\prime}\) the sum over integers \(k\) with \( p_1 \nmid k \text{ and } \cdots \text{ and } p_j \nmid k\)
• \( x \ge p_1 p_2 \cdots\ p_j \)

An example
I calculate an example for \( x = 50 \) and \( P = \{ 2,3 \} \). The first term is
\begin{equation}
\sum_{d \le 7}\!\phantom{}^{\prime}\ M\left( \frac{50}{d} \right) = M(50) + M(10) + M(7)
\end{equation}
The second term is
\begin{equation}
\sum_{d \le 7} \mu(d) \omega_P\left( \frac{50}{d} \right)
\end{equation}
If I use
\begin{equation}
\omega_P(x) = \lfloor x \rfloor - \left\lfloor \frac{x}{2} \right\rfloor -  \left\lfloor \frac{x}{3}\right\rfloor + \left\lfloor \frac{x}{6} \right\rfloor
\end{equation}
then it can be calculated that the second term is zero. The third term is \( \omega(7) M(7) = 3 M(7) \). All together, von Sterneck's formula gives the recursive relation
\begin{equation}
M(50) + M(10) +M(7) + 0 - 3 M(7) =0
\end{equation}
Because \( M(10) = -1 \) and \( M(7) = -2 \) this gives \( M(50) = -3 \).

Sketch of the proof of the formula
The proof proceeds as follows. Define the function \( e_P \)
\begin{equation}
e_P(m) = \begin{cases} 1 \quad\text{if}\quad p_1 \nmid m \text{ and } \cdots \text{ and } p_j \nmid m\\ 0 \quad\text{otherwise} \end{cases}
\end{equation}
then calculate the Dirichlet convolution 

\begin{equation} e_P \ast\mu = 1_{\langle P \rangle} \cdot \mu
\end{equation} where \( 1_{\langle P \rangle}\) is the characteristic function on the set \( \langle P \rangle = \left\{ n \in \mathbb{N} \middle|\quad \forall p: p | n \implies p \in P \right\} \). Because \( \sum_{ k \le x}e_P(k) = \omega_P(x) \), Dirichlet's hyperbola method gives
\begin{equation}
\sum_{n \le x} 1_{\langle P \rangle} (n) \mu(n) = \sum_{d \le \sqrt{x}}e_P(d) \ M\left( \frac{x}{d} \right) + \sum_{d \le \sqrt{x}} \mu(d) \omega_P\left( \frac{x}{d} \right) - \omega_P(\sqrt{x})M(\sqrt{x})
\end{equation}
The result then follows because the left hand side is zero for \( x \ge p_1 p_2 \cdots\ p_j \)

Von Sterneck used this formula with \( P = \{ 2,3,5,7 \} \). In that way he could calculate values of \(M(n) \)  for \( n \) up to 5,000,000 based on a pre-computed table of \(M(n) \)  for \( n \) up to 500,000.

Neue Empirische Daten Über Die Zahlentheoretische Funktion sigma(n), von Sterneck, 1913. Part I

In the paper in the title, von Sterneck performs numerical tests on the inequality
\begin{equation}\label{Neueeq:1}
| M (n) | \le \sqrt{n}
\end{equation}
where \( M(n) = \displaystyle\sum_{k=1}^n \mu(n) \) is the Mertens function. The inequality (1) is of historical interest, because it implies the correctness of the Riemann hypothesis. In 1985 however, Odlyzko and te Riele, showed that (1) is incorrect, although an explicit number \( n \) for which (1) fails is still not known. Nevertheless, I find von Sterneck's paper still interesting because I want to see how \( M(n) \) was calculated for large values of \( n \) before the advent of (electronic) computers.