\newtheorem macro takes the following arguments:
where you can leave out the arguments in square brackets, i.e.
[numberby]. These arguments have the following meaning:
counter gives the name of an existing counter that should be used for numbering the theorems. A typical use-case is that you want to have a continous numbering that includes both equations and theorems, or different types of theorems. However, you cannot use this argument to specify the style of the numbering of the equation.
numberby gives the name of an existing counter that should be used as a parent number to the theorem's number. A typical use-case is that you want to have a continous numbering of theorems only inside each chapter or section. That's the case here: You want to have numbers like "5.1" meaning "theorem 1 of chapter 5".
Printed output should not be empty because it inserts additional space even if it is empty. Use whatever label you want to give to your theorems (important distinction: the counters only count, i.e. they only take numbers; everything else has to go to the label).
What you can do in your specific case is: Choose
\S (which results in §) as
Printed output in your theorem definition, and then set the value of the
section counter to 5:
Something here again