THE LEAST QUADRATIC NONRESIDUE AND VINOGRADOV'S HYPOTHESIS.
Keywords:
equidistribution in arithmetic progressions, dispersion method.Abstract
Let αm and βn be two sequences of real numbers supported on [M,2M] and [N,2N] with M = X1/2−δ and N = X1/2+δ. We show that there exists a δ0 > 0 such that the multiplicative convolution of αm and βn has exponent of distribution (in a weak sense) as long as 0 ≤ δ < δ0, the sequence βn is Siegel-Walfisz and both sequences αm and βn are bounded above by divisor functions. Our result is thus a general dispersion estimate for “narrow” type-II sums. The proof relies crucially on Linnik’s dispersion method and recent bounds for trilinear forms in Kloosterman fractions due to Bettin-Chandee. We highlight an application related to the Titchmarsh divisor problem.
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