See e.g. Theorem 12.6 of Iwaniec-Kowalski.

]]>If is the Fourier transform of , then is the Fourier transform of , as can be seen by a simple change of variables .

]]>“Note that if is coprime to , then is a scalar multiple of by a quantity of magnitude ; taking Fourier transforms, this implies that and also differ by this factor. “

]]>I’m asking myself, too, if there is a way to deeper exploit quadratic reciprocity.

By assuming the independence of the values taken by the Legendre symbol modulus q over primes, one would expect to have bounded by some (potentially huge) power of .

Moreover, if and, for some prime ,

, then the prime number is a quadratic non-residue modulus q.

Is there a way to exploit the Lovasz Local Lemma in the last setting?

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