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The question is pretty much in the title. Euler's Totient function $\varphi(n)$ satisfy the following formula:

$$\varphi(n) =n \prod_{p|n}\left(1-\frac{1}{p}\right)$$

Is it possible through this formula or something else to represent $\varphi(n)$ as an integral?

Casimir Rönnlöf
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1 Answers1

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One possible way is to consider the Dirichlet series of the Euler's totient function, and arrive at an integral using the coefficient inversion formula (relevant content: sections 2 and 7 here: https://en.wikipedia.org/wiki/Dirichlet_series).

I do not know if this satisfies the conditions in the question asked (because it was slightly vague) but I hope it helps.

something
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