Du
Shedule -
Rencontres de théorie analytique des nombres
Large sieve inequalities for exceptional Maass forms and applications
Salle Grisvard, IHP, Paris
A number of results on classical problems in analytic number theory rely on bounds for multilinear forms of Kloosterman sums, which in turn use deep inputs from the spectral theory of automorphic forms. We’ll discuss our recent work available at <a href="https://arxiv.org/abs/2404.04239"> arxiv.org/abs/2404.04239</a>, which uses this interplay between counting problems, exponential sums and automorphic forms to improve results on the greatest prime factor of $n^2+1$, and on the exponents of distribution of primes and smooth numbers in arithmetic progressions.
The new ingredients are certain large sieve inequalities for exceptional Maass forms, which improve on pioneering results of Deshouillers-Iwaniec in special settings. These act as on-average substitutes for Selberg’s eigenvalue conjecture, narrowing (and sometimes completely closing) the gap between previous conditional and unconditional results.