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In this paper, we evaluate a smoothed sum of the form $\displaystyle\sideset{}{^*}\sum_{d \leq X}\left(\sum_{n \leq Y} \left(\frac{8d}{n} \right )\right)^2$, where $\left ( \frac{8d}{\cdot} \right )$ is the Kronecker symboland $\sideset{}{^*}\sum$ denotes a sum over positive odd square-free integers.

The mean square of real character sums