The present paper is a short survey on some of the older work (before the 1960’s) done on the function \(\Psi(x,y)\) which counts the number of positive integers \(n\leq x\) free of primes \(p>y\). It starts with Dickman’s 1930 result [K. Dickman, Ark. Mat. Astron. Fys. 22 A, No. 10, 14 p. (1930; JFM 56.0178.04)] to the effect that \(\rho(u)=\lim_{x\to\infty} \Psi(x,x^{1/u})/x\) exists for every fixed \(u>0\) and continues with de Bruijn’s estimate on the error term \(\Psi(x,x^{1/u})/x-\rho(u)\). Various results are presented on the Dickman-de Bruijn function \(\rho(u)\). The paper continues by presenting contributions to the topic by each of S. Ramanujan, I. M. Vinogradov, K. Dickman, S. S. Pillai, R. A. Rankin, A. A. Buchstab, V. Ramaswami and S. Chowla.
On a personal note, this reviewer remarks that the author has chosen to use neither the North American terminology of smooth numbers nor his own terminology of psixyology (for the theory of the function \(\Psi(x,y)\)) in favor of the French terminology “friable” as the property of a positive integer \(n\) to be free of large primes.