most recent 30 from http://mathoverflow.net 2013-05-21T22:23:10Z http://mathoverflow.net/feeds/question/91718 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91718/linear-combination-of-multiplicative-functions Charles 2012-03-20T14:10:58Z 2012-03-20T14:10:58Z

<p>Carlitz showed necessary and sufficient conditions for an arithmetic function to be a linear combination of two multiplicative functions. He mentions the possibility of generalizing to $k$ multiplicative functions, but as far as I can tell this was never published.</p> <p>What is known about arithmetic functions which can be represented as linear combinations of some fixed number $k$ of multiplicative functions? Given $k$, how many terms 1, 2, ..., n are needed to either reject it as a linear combination of $k$ multiplicative functions or to generate (partial) multiplicative functions and their coefficients?</p> <p>[1] L. Carlitz, <a href="http://www.collectanea.ub.edu/index.php/Collectanea/article/viewArticle/3347" rel="nofollow">Sums of arithmetic functions</a>, <em>Collectanea Mathematica</em> <strong>20</strong>:2 (1969), pp. 107-126.</p>