Real zeros of holomorphic Hecke cusp forms and sieving short intervals

Abstract

<p> Abstract. We study so-called real zeros of holomorphic Hecke cusp forms,<br /> that is zeros on three geodesic segments on which the cusp form (or a multiple<br /> of it) takes real values. Ghosh and Sarnak, who were the first to study this<br /> problem, showed that existence of many such zeros follows if many short intervals<br /> contain numbers whose all prime factors belong to a certain subset of<br /> the primes. We prove new results concerning this sieving problem which leads<br /> to improved lower bounds for the number of real zeros.</p>