When I was doing some task of analytic number theory I was stuck on computing this sum $$S:=\frac{1}{L} \sum_{q \in \mathcal{Q}} \phi(q) \overline{a}^{\frac{1}{2}},$$ where $\overline{a}$ is the inverse of $a$ modulo $q>0,$ $\mathcal{Q}$ is a nonempty set of numbers defined by $$\mathcal{Q}=\left\{q \in [Q,2Q]; \ \gcd(a,q)=1\right\}, \quad Q\geq 1$$ and $$L=\sum_{q \in \mathcal{Q}} \phi(q),$$ $\phi$ is the Euler function. Can someone help me in doing this task? Any help is appreciated.
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1$\begingroup$ it's not clear what range for $\bar{a}$ you are expecting. If we just assume that it is in the range of $(0,q)$, I think you can simply consider a function $H: \mathbb{R} \to \mathbb{R}$, where $H(n) = n^{1/2}$ for integers in the range $[0,2Q]$ and 0 elsewhere. Then $\bar{a}^{1/2}$ mod $q$ is merely $\sum_{x \in \mathbb{N}} H\left(\frac{ax-1}{q}\right)$, up to some small error. You can clearly smooth this function as well. $\endgroup$– PigDec 13, 2015 at 19:22
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$\begingroup$ Just to make sure: you really want the real, positive square root of $a\in(0,q)$, not the mod $q$ version of that? $\endgroup$– Fan ZhengDec 13, 2015 at 20:20
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$\begingroup$ @FanZheng, I want the square root of the inverse of a fixed integer $a$ modulo $q.$ $\endgroup$– Khadija MbarkiDec 13, 2015 at 20:43
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1$\begingroup$ @KhadijaMbarki Then how do you make sure that $a$ is a quadratic residue mod $q$? $\endgroup$– Fan ZhengDec 13, 2015 at 21:10
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1$\begingroup$ @KhadijaMbarki Also, you are summing over residue classes in different moduli, which doesn't make much sense, unless you specify a representative of that residue class. $\endgroup$– Fan ZhengDec 13, 2015 at 21:11
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