Real zeros of holomorphic Hecke cusp forms and sieving short intervals

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