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A result of Heath-Brown states: For $a_1,...,a_n$ be arbitrary complex numbers, $$\sideset{}{^*}\sum_{m\le M} \left|\sideset{}{^*}\sum_{n\le M}a_n\left(\frac{n}{m}\right)\right|^{2} \ll_{\epsilon}(MN)^{\epsilon}(M+N)\sideset{}{^*}\sum_{n\le N}|a_n|^2$$ $\sideset{}{^*}\sum$ indicates restriction to odd square free values.

Is the implied constant effective ?

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    $\begingroup$ Do you want to give a specific reference to this result? $\endgroup$
    – Greg Martin
    Commented Apr 8, 2019 at 17:16

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First, for reference, the estimate comes from

"A mean value estimate for real character sums", Acta Arithmetica, 72, 1995, 235--275.

From a quick look, I think that the constant should be effective, but certainly not explicit (the proof is quite complicated; if I remember correctly, there is another paper in which Heath-Brown refers to the possibility of having to adapt this result to another situation as being "distinctly unpleasant", or words to that effect). It shouldn't be too hard to check that this is indeed effective by following the argument without checking all details, since the proof has very few -- if any -- outside references from which ineffectiveness might arise.

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