Most PhD positions in Germany are regular employment positions, and they are offered whenever a vacancy arises, i.e. there is no recurring deadline. Since those positions are often (sometimes must be) advertised, you should monitor relevant job bulletins. I'm not a mathematician, but I imagine the German Mathematical Society or similar groups have a newsletter or liststerv over which job postings are distributed, for example.
Some PhD positions in Germany are funded by stipends, in particular those at Graduate Schools (less common than in the US), but there are also other foundations that may offer stipends for PhD students. In both cases, positions are offered periodically, namely twice per year (once per semester), mostly in spring and fall. For example, the BIGS graduate school invites applications until 30 November and 30 April each year.