Mertens function

In number theory, the Mertens function is defined for all positive integers n as:

$$ M(n) = \sum_{k=1}^{n} \mu(k) $$

where $\mu(k)$ is the Möbius function. The function is named in honor of the German mathematician Franz Mertens (1840–1927). The definition can be extended to positive real numbers by $M(x) = M(\lfloor x \rfloor)$.

Less formally, $M(x)$ is the count of square-free integers up to $x$ that have an even number of prime factors, minus the count of those that have an odd number.

Properties and Significance

Because the Möbius function only takes the values −1, 0, and +1, the Mertens function moves slowly, and there is no $x$ such that $|M(x)| > x$. H. Davenport demonstrated that for any fixed $h$,

$$ M(x) = O\left(\frac{x}{\log^h x}\right). $$

The Mertens conjecture stated that $|M(x)| < \sqrt{x}$ for all $x > 1$. This conjecture was proven false in 1985 by Andrew Odlyzko and Herman te Riele. However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of $M(x)$, namely $M(x) = O(x^{1/2 + \varepsilon})$.

Representations

The Mertens function has several important representations:

• As an integral using the Riemann zeta function: $\displaystyle \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{x^s}{s\zeta(s)},ds = M(x)$, where $c > 1$.
• As a determinant of the $n \times n$ Redheffer matrix, a (0,1)-matrix in which $a_{ij}$ is 1 if either $j = 1$ or $i$ divides $j$.
• As a sum over Farey sequences: $M(n) = -1 + \sum_{a \in \mathcal{F}_n} e^{2\pi i a}$, where $\mathcal{F}_n$ is the Farey sequence of order $n$.

Calculation

The Mertens function has been computed for increasingly large ranges using sieve methods. Notable computational milestones include:

A combinatorial algorithm developed incrementally by Meissel, Lehmer, Lagarias–Miller–Odlyzko, and Deléglise–Rivat computes isolated values of $M(x)$ in $O(x^{2/3}(\log\log x)^{1/3})$ time. A further improvement by Harald Helfgott and Lola Thompson in 2021 achieves $O(x^{3/5}(\log x)^{3/5+\varepsilon})$ time.

Known Upper Bounds

Assuming the Riemann hypothesis, Ng notes that $M(x) = O\left(\sqrt{x} \exp\left(\frac{C \cdot \log x}{\log\log x}\right)\right)$ for some positive constant $C > 0$. Without assuming the RH, explicit bounds such as $|M(x)| < 0.6437752 \cdot x / \log x$ for $x > 1$ are known.

See Also

• Möbius function
• Mertens conjecture
• Riemann hypothesis
• Redheffer matrix
• Farey sequence