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How does one use Euler Maclaurin to compute asymptotics for sums like $$ \sum_{\substack{n\le x \\ (n,q)=1}} \frac{1}{\sqrt{n}} \quad \text{or} \quad \sum_{\substack{n\le x \\ (n,q)=1}} \frac{\log n}{\sqrt{n}}?$$

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You can get asymptotics using Abel summation (see Tenenbaum's Introduction to Analytic and Probabilistic Number Theory, section 1.0).

We can let $b(x)=\frac{1}{\sqrt{x}}$ and define $a_n$ to be $1$ if $(n,q)=1$ and 0 otherwise. Then $$ \sum_{\substack{n\le x \\ (n,q)=1}} \frac{1}{\sqrt{n}} =\sum_{n \le x} a_n b(n) =A(x)b(x) - \int_1^x A(t) b'(t) \, dt = \frac{A(x)}{\sqrt{x}} + \frac{1}{2} \int_1^x A(t) t^{-3/2} \, dt $$ where $$ A(x) = \sum_{n \le x} a_n = \frac{x}{q} \phi(q) + O(\phi(q)).$$ From this, we can determine that $$\sum_{\substack{n\le x \\ (n,q)=1}} \frac{1}{\sqrt{n}} = 2 \sqrt{x} \frac{\phi(q)}{q} + O( \phi(q) ). $$ I am sure better bounds on the error terms are possible, but this might be good enough for what you want to do with it.

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We can actually get an additional term using the Wiener-Ikehara theorem.

Introduce the Dirichlet Series $A(s)$ whose terms are given by the indicator $(n, q) =1$ times $1/\sqrt{n}$. We have $$ A(s) = \sum_{(n,q)=1} \frac{1/\sqrt{n}}{n^s} = \sum_{(n,q)=1} \frac{1}{n^{s+\frac{1}{2}}} = \prod_{p\nmid q} \frac{1}{1-\frac{1}{p^{s+\frac{1}{2}}}} = \zeta\left(s+\frac{1}{2}\right) \prod_{p\mid q} \left(1-\frac{1}{p^{s+\frac{1}{2}}} \right),$$ where $p$ ranges over the primes. Furthermore, introduce $$ B(s) = A(s) - \zeta\left(\frac{1}{2}\right) \prod_{p\mid q} \left(1-\frac{1}{\sqrt{p}}\right)$$ This Dirichlet series differs from $A(s)$ in its constant term and converges in $\mathfrak{R}(s) \ge \frac{1}{2}.$ It has a simple pole at $\frac{1}{2}$ and is zero at $s=0.$ Wiener-Ikehara applies to $B(s)$, giving

$$\sum_{k\le n,(k,n)=1} \frac{1}{\sqrt{k}} - \zeta\left(\frac{1}{2}\right) \prod_{p\mid q} \left(1-\frac{1}{\sqrt{p}}\right) \sim \prod_{p\mid q} \left(1-\frac{1}{p}\right) \frac{\sqrt{n}}{1/2} = 2 \frac{\phi(q)}{q} \sqrt{n}.$$ We construct the zero at $s=0$ because we are actually working with the Mellin-Perron type integral $$\int_{1-i\infty}^{1+i\infty} B(s) n^s \frac{ds}{s}$$ and need to cancel the pole at $s=0$.

The conclusion is that $$ \sum_{k\le n,(k,n)=1} \frac{1}{\sqrt{k}} \sim 2 \frac{\phi(q)}{q} \sqrt{n} + \zeta\left(\frac{1}{2}\right) \prod_{p\mid q} \left(1-\frac{1}{\sqrt{p}}\right)$$ The numerics of this approximation are excellent even for small values of $n$.

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