Mertens function

This article is about a function defined on positive reals (and in particular, natural numbers) obtained as the summatory function of an arithmetic function, namely Mobius function.
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Definition

Let $x$ be a positive real number. The Mertens function of $x$, denoted $M(x)$, is defined as:

$\sum _{n\leq x}\mu (n)$.

Here, $\mu$ denotes the Mobius function.

We typically evaluate the Mertens function for natural numbers.