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.