If you have two threads invoking a static function at the same moment in time, is there a concurrency risk? And if that function uses a static member of the class, is there even
Yes, there is a risk. That's why you'll see in MSDN doc, it will often say "This class is threadsafe for static members" (or something like that). It means when MS wrote the code, they intentionally used synchronization primitives to make the static members threadsafe. This is common when writing libraries and frameworks, because it is easier to make static members threadsafe than instance members, because you don't know what the library user is going to want to do with instances. If they made instance members threadsafe for many of the library classes, they would put too many restrictions on you ... so often they let you handle it.
So you likewise need to make your static members threadsafe (or document that they aren't).
By the way, static constructors are threadsafe in a sense. The CLR will make sure they are called only once and will prevent 2 threads from getting into a static constructor.
EDIT: Marc pointed out in the comments an edge case in which static constructors are not threadsafe. If you use reflection to explicitly call a static constructor, apparently you can call it more than once. So I revise the statement as follows: as long as you are relying on the CLR to decide when to call your static constructor, then the CLR will prevent it from being called more than once, and it will also prevent the static ctor from being called re-entrantly.