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$\displaystyle \begin{aligned} S(x, \alpha) &= \sum_{n \le x} \Lambda(n) e(na/q) e(n\beta) \\ &= \sum_{n \le x} e(n\beta) \left( S(n, a/q) - S(n-1, a/q) \right) \\ &= \sum_{n \le x} e(n\beta) \left( \frac{\mu(q)}{\phi(q)} + \text{Err}(n, \alpha) - \text{Err}(n-1, \alpha) \right) \\ &= \frac{\mu(q)}{\phi(q)} \left( \sum_{n \le x} e(n\beta) \right) + \sum_{1 \le m \le x-1} \left( e( (m+1)\beta) - e( m\beta ) \right) \text{Err}(m, \alpha) \\ &\qquad + e(x\beta) \text{Err}(x, \alpha) - e(0) \text{Err}(0, \alpha) \\ &\le \frac{\mu(q)}{\phi(q)} \left( \sum_{n \le x} e(n\beta) \right) + \left( \sum_{1 \le m \le x-1} \|\beta\| \text{Err}(m, \alpha) \right) + \text{Err}(0, \alpha) + \text{Err}(x, \alpha) \\ &\ll \frac{\mu(q)}{\phi(q)} \left( \sum_{n \le x} e(n\beta) \right) + \left( 1+x\left\| \beta \right\| \right) O\left( \sqrt{qx} (\log qx)^2 \right) \end{aligned}$

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