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 the existence of many such zeros if many short intervals contain numbers whose prime factors all 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.

Soient $\lambda \>1,0\<\eta \<\frac{1}{2}$ et $g\left(n\right)$ une fonction multiplicative vérifiant $\left\{\begin{array}{c}g\left(p\right)=1/\lambda \\ g\left(n\right)\gg n^{-\eta }\end{array}\right.$. Dans ce travail, on établit une formule asymptotique de la somme ${\sum }_{ng\left(n\right)\le x;P\left(n\right)\le y}1$, valable dans le domaine exp${(loglogcx)}^{\frac{5}{3}+ϵ}\le y/\lambda \le cx$, et on donne une condition nécessaire et suffisante pour que cette somme soit équivalente à ${\sum }_{n\le x;P\left(n\right)\le y}1/g\left(n\right)$.