An arithmetic function
ƒ( n) is said to be additiveif it satisfies ƒ(ab) = ƒ(a)+ ƒ(b)whenever aand bare coprime integers. For such a function we define
A standard form of the Turán-Kubilius inequality states that
holds for some absolute constant
c1, uniformly for all complex-valued additive arithmetic functions ƒ( n), and real x≧ 2. An inequality of this type was first established by Turán [ 11], [ 12] subject to some side conditions upon the size of │ƒ(pm)│.For the general inequality we refer to [ 10].
This inequality, and more recently its dual, have been applied many times to the study of arithmetic functions. For an overview of some applications we refer to [
2]; a complete catalogue of the applications of the inequality (1) would already be very large. For some applications of the dual of (1) see [ 3], [ 4], and [ 1].;