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The Ramanujan sum is $c_n(k)={\sum }_{1\le j\le n,(j,n)=1}{e}^{2\pi ijk/n}$. It is an integer: insert $1_{(j,n)=1}={\sum }_{d\mid (j,n)}\mu (d)$, sum the geometric series, and obtain $c_n(k)={\sum }_{r\mid (n,k)}r\mu (n/r)$. Here $\mu$ is the Möbius function. Such sums connect roots of unity with integer character of a representation values.

A Type I sum in analytic number theory has the form ${\sum }_{d\le U}{a}_d{\sum }_{m\in I_d}f(dm)$, with a short weighted variable and a relatively long inner sum whose coefficients are simple. The Vaughan identity produces these sums when the Möbius function is truncated.