High-Power Analogues of the Turán-Kubilius Inequality, and an Application to Number Theory

Elliott, P. D. T. A.

Canadian journal of mathematics, Tome 32 (1980) no. 4, pp. 893-907

Résumé

An arithmetic function ƒ(n) is said to be additive if it satisfies ƒ(ab) = ƒ(a) + ƒ(b) whenever a and b are coprime integers. For such a function we define A standard form of the Turán-Kubilius inequality states that(1) holds for some absolute constant c 1, 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].

Détail

DOI : 10.4153/CJM-1980-068-0

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