Bertfried Fauser <[email protected]> writes: > 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).
Unfortunately, there is a "slight" problem with testing for such properties: it's not possible. There is no way to test whether a given function is always zero. (Somewhat related: there is no way to get an InputForm for an element of DIRRING.) However, it would be very easy to have a domain of multiplicative functions, by simply requiring a definition for prime number arguments only. I do think it would be nice to provide also a domain for Dirichlet series f(s) = sum f_n/n^s. I guess in this case a representation as stream would indeed be more natural. Martin -- 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.
