Stability problem for number-theoretically multiplicative functions
HTML articles powered by AMS MathViewer
- by Tomasz Kochanek and Michał Lewicki PDF
- Proc. Amer. Math. Soc. 135 (2007), 2591-2597 Request permission
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.References
- John Baker, J. Lawrence, and F. Zorzitto, The stability of the equation $f(x+y)=f(x)f(y)$, Proc. Amer. Math. Soc. 74 (1979), no. 2, 242–246. MR 524294, DOI 10.1090/S0002-9939-1979-0524294-6
- John A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), no. 3, 411–416. MR 580995, DOI 10.1090/S0002-9939-1980-0580995-3
Additional Information
- Tomasz Kochanek
- Affiliation: Institute of Mathematics, Silesian University, Bankowa 14, PL-40 007 Katowice, Poland
- MR Author ID: 811694
- Email: t_kochanek@wp.pl
- Michał Lewicki
- Affiliation: Institute of Mathematics, Silesian University, Bankowa 14, PL-40 007 Katowice, Poland
- Email: m_lewicki@wp.pl
- Received by editor(s): May 1, 2006
- Published electronically: February 9, 2007
- Communicated by: Jonathan M. Borwein
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 2591-2597
- MSC (2000): Primary 39B82
- DOI: https://doi.org/10.1090/S0002-9939-07-08854-5
- MathSciNet review: 2302580