My fault, sorry. So section specific templates lie under the layouts/section directory. And the paginator works fine in the section's template, so my problem is solved.
Just one more thing. I tired and I put some content file into the content/series directory, so now they can be reached on the "series/content-file-name" URL which is fine, but in the same time I also have a paginator for the series section so the "series/page/1", "series/page/2", etc URLs are also valid. It seems that both the pager and the individual content pages can be reached fine, but I guess if I would have a "page" named sub-directory with some content file in content/series that would broke the system. But until that, this should work fine I guess.