Stability problem for number-theoretically multiplicative functions

Abstract:

We deal with the stability question for multiplicative mappings in the sense of number theory. It turns out that the conditional stability assumption: \[ |f(xy)-f(x)f(y)|\leq \varepsilon \;\text {for relatively prime $x$, $y$} \] implies that $f$ lies near to some number-theoretically multiplicative function. The domain of $f$ can be general enough to admit, in special cases, the reduction of our result to the well known J. A. Baker - J. Lawrence - F. Zorzitto superstability theorem.