In number theory, the Ramanujan sum function is defined as below: $$c_q(n)=\sum_{\substack{1\le k\le q\\ \gcd(k,q)=1}}\exp\left(\frac{2\pi i k n}{q}\right)$$ where $q,n$ are integers. This Wikipedia article shows a list of amazing properties of this arithmetic function. I was just wondering,
Is there a generalization for this sum for real $n$? That is, if I define a function as $$c_q(r)=\sum_{\substack{1\le k\le q\\ \gcd(k,q)=1}}\exp\left(\frac{2\pi i k r}{q}\right)$$ where $r$ is a real number, is there any mention in the literature of such a function and some of its properties?
I searched a while but could not find any mention of such a function, and I am also not sure how to find properties of this function (that is properties equivalent to the case when $n$ is an integer) when $n$ is real (I pretty much could derive most of the properties when $n$ is an integer). For a specific example, when, $n$ is a positive integer, the Wikipedia article shows that there is an explicit formula for $c_q(n)$, found by Holder, $$c_{q}(n)=\mu\left(\frac{q}{\gcd(q,n)}\right)\frac{\phi(q)}{\phi\left(\frac{q}{\gcd(q,n)}\right)}$$ Is there such a formula available for the function when $n$ is real? Any help with references, or some ideas will be helpful. Thanks in advance.