Combining sieve methods with automorphic form theory and techniques from -adic cohomology, we prove that the sign of Kloosterman sums Kl(1, 1; n) changes infinitely often as n ranges over the squarefree integers having all their prime factors larger than n1/ . 1. Introduction Soient a, b et n trois entiers, avec n 1. On rappelle que la somme de Kloosterman Kl(a, b; n) est d´finie par la formule e ax + bx Kl(a, b; n) = exp 2πi . n x mod n (x,n)=1 (la notation x indique l’inverse de x modulo n). Rappelons que c’est un nombre r´el, qui,.