8 May
2019
8 May
'19
5 p.m.
Can every suitably nice function A(s) be "inverted" to give an arithmetic function a(n) such that A(s) = sum_{n >= 1} a(n)/n^s? (E.g. by Perron's formula?) In other words, given ordinary generating functions A_i(x) = sum_{n >= 0} a^i_n x^n, I can combine them into a Dirichlet generating function A(s) = prod_{i >= 1} A_i(p_i^{-s}). My question is whether given a suitably nice but otherwise arbitrary function of s and think of it as a Dirichlet generating function, can I decompose it into a collection of ordinary generating functions? If so, what constitutes "suitably nice"? Merely convergence for Re(s) > sigma? -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com