Gaps between norm forms and related questions

Mardi 30 janvier 2018 14:15-15:15 - Maksym Radziwill - McGill

Résumé : A question of interest in analytic number theory is the study of gaps between sequences that are multiplicative in nature, e.g primes, sums of two squares, or more generally norm-forms. In this direction an old conjecture of Erdos predicts the order of magnitude of the second moment of gaps between primes. This conjecture generated further works for the sequence of almost primes (Friedlander) and sums of two squares (Hooley). We will focus on the result of Hooley and specifically on the question whether his result can be extended to norm-forms of number fields of degree exceeding 2. So far such a generalization has resisted all attempts. The reason is that the natural extension of Hooley’s approach requires the solution of a shifted convolution problem for coefficients of L-functions of degree exceeding two. The latter is an outstanding problem in analytic number theory. Despite these apparent difficulties I will describe recent work with Matomaki in which we establish a generalization of Hooley’s result to all norm forms. I will place special emphasis on how the core difficulties of the shifted convolution problem are avoided.

Lieu : IMO Bât. 307, salle 3L15

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