Totient summatory 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 Euler phi-function.
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Definition

Let $x$ be a positive real number. The totient summatory function of $x$ is defined as:

$\Phi (x)=\sum _{n\leq x}\varphi (n)$

where $\varphi$ is the Euler phi-function.