ACTA ARITHMETICA 146.3 (2011) Exponential sums involving the largest prime factor function by Jean-Marie De Koninck (Qu´ebec) and Imre K´ atai (Budapest) 1. Introduction. Let P (n) stand for the largest prime factor of the integer n ≥ 2 and set P (1) = 1. A well known result of I. M. Vinogradov [7] asserts that, given any irrational number α, the sequence αp n , n =1, 2,..., where p n stands for the nth prime, is uniformly distributed in [0, 1]. In 2005, Banks, Harman and Shparlinski [1] proved that for every irrational number α, the sequence αP (n), n =1, 2,..., is uniformly distributed mod 1. They did so by using the well known Weyl criteria (see the book of Kuipers and Niederreiter [5]) and thus by establishing that (1.1) lim x→∞ 1 x X n≤x e(αP (n)) = 0, where e(z ) := exp{2πiz }. Let M stand for the set of all complex-valued multiplicative functions and let f M be the subset of those functions f ∈M such that |f (n)|≤ 1 for positive integers n. Daboussi (see Daboussi and Delange [2]) proved that given f ∈ f M and any irrational number α, lim x→∞ sup f ∈ f M 1 x X n≤x f (n)e(nα)=0. Let M 1 stand for the subset of those functions f ∈M such that |f (n)| =1 for all positive integers n. In this paper, we first generalize (1.1) by showing that for any irrational number α and any function f ∈M 1 , we have (1.2) X n≤x f (n)e(αP (n)) = o(x) (x →∞). We also show that this general result further holds if one replaces e(αP (n)) by T (P (n)), where T is any function defined on primes satisfying |T (p)| =1 2010 Mathematics Subject Classification : 11N13, 11L07. Key words and phrases : largest prime factor, exponential sums, uniform distribution. DOI: 10.4064/aa146-3-3 [233] c  Instytut Matematyczny PAN, 2011