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Classical versions of Kolmogorov complexity are incomputable. Nevertheless,in 1975 Solovay showed that there are computable functions $f > K+O(1)$ suchthat for infinitely many strings $\sigma$, $f(\sigma)=K(\sigma)+O(1)$, where$K$ denotes prefix-free Kolmogorov complexity (while $C$ denotes plainKolmogorov complexity). Such an $f$ is now called a Solovay function. We provethat many classical results about $K$ can be obtained by replacing $K$ by aSolovay function. For example, the three following properties of a function $g$all hold for the function $K$.  (i) The sum of the terms $\sum_n 2^{-g(n)}$ is a Martin-L\"of random real.  (ii) A sequence A is Martin-L\"of random if and only if $C(A \upharpoonrightn) > n -g(n)-O(1)$.  (iii) A sequence A is K-trivial if and only if $K(A \upharpoonright n) < g(n)+ O(1)$.  We show that when fixing any of these three properties, then among allcomputable functions exactly the Solovay functions possess this property.Furthermore, this characterization extends accordingly to the larger class ofright-c.e. functions.

Solovay functions and their applications in algorithmic randomness