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A CHARACTERISATION THEOREM OF THE LOGARITHMIC FUNCTION MODULO 1

The original idea of such characterisations is that of [2]. He proved that if a real-valued additive function is non-decreasing, or satisfies f(n + 1) − f(n) → 0 (as n → ∞), then it must have the form c log n for some constant c. He had a separate argument for each case. In [1] Elliott solves the problem in a generalised form: Let a > 0, b, A > 0 and B be integers with ∆ = aB − Ab non-zero. If G is a real additive arithmetic function and for some constant C satisfies G(an + b)−G(An + B) → C as n →∞, then there is a further constant c such that G(x) = c log x for every x ∈ N which are prime to aA∆. The problem can be generalised further, there is a great variety of conditions that can be introduced giving numerous different results. This paper gives proof of the following rather interesting theorem: