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Recently I have been familiarized with the concept of 'Möbius function'. As far I
know the Möbius function is defined by

$$\mu(n) = \begin{cases} 1 & \small \text{if $n$ is a square-free positive integer with an even number of prime factors.} \\ -1 & \small \text{if $n$ is a square-free positive integer with an odd number of prime factors.}\\ 0 & \small \text{if n has a squared prime factor.} \end{cases} $$

But I'm very interested to know how the concept of Möbius function developed historically. What situations or problems led the mathematicians to initiate the concept of Möbius function? Can anyone explain?

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See https://en.wikipedia.org/wiki/Möbius_function

The German mathematician August Ferdinand Möbius introduced it in 1832.

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Hardy & Wright, Notes on ch. XVI: "... μ(n) occurs implicitly in the works of Euler as early as 1748, but Möbius, in 1832, was the first to investigate its properties systematically."

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In the Disquisitiones Arithmeticae (1801) Carl Friedrich Gauss showed that the sum of the primitive roots (mod p) is μ(p − 1), (see #Properties and applications) but he didn't make further use of the function. In particular, he didn't use Möbius inversion in the Disquisitiones.

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