Hi Martin, > Anybody knows of any additional structure this ring carries? I guess I > should make it an Algebra Coef, right?
This ring has many nice properties, especially if seen as ring over the complex numbers, that is f : N --> C, see for example: http://arxiv.org/abs/math-ph/0511079 I would especially like to see the Hopf algebra structure on this ring, which works on the coefficients. Note that the convolution product of f,g is (f*g)(n) = \sum_{d|n} f(d)g(n/d) from which you can read of the coproduct \Delta(n) -> \sum_{d|n} [d , n/d] where [d,n/d] is an ordered pair in N x N on which functions N x N --> C live, I am not sure if Franz' tensor package would help here. Moreover some tests would be nice, eg if an arithmetic function is: a) completely multiplicative b) multiplicative c) non of a) or b) You also might like to have Dirichlet characters \chi, that are arithmetic functions with a periodicity property, that is \chi(n)=\chi(n+k) for some k >0 in N. etc... You might test for further properties, as providing a abscissa of absolute convergence (each arithmetic function has one). I would like to have, but don't see how one could implement such things features like: i) given an arithmetic f, is there a functional equation, and if so what does it look like. ii) provide the analytic continuation of an arithmetic function extending it as a meromorphic function on the whole of C-{set of poles} iii) provide the set of poles of an arithmetic function (possibly as a zero set of a polynomial ideal) vi) etc... I will certainly has soon a deeper look at this. Cheers BF. -- % PD Dr Bertfried Fauser % Research Fellow, School of Computer Science, Univ. of Birmingham % Honorary Associate, University of Tasmania % Privat Docent: University of Konstanz, Physics Dept <http://www.uni-konstanz.de> % contact |-> URL : http://www.cs.bham.ac.uk/~fauserb/ % Phone : +44-121-41-42795 -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/fricas-devel?hl=en.
