phd - What is expected in a masters thesis of a mathematics student?

What is the level of work expected in the masters thesis of a student of maths?

I know some people's works are worthy of publications while some involve only studying some topic in detail from a book and submitting a summary (since this is akin to a couple of courses, a year worth of work, in that topic in terms of content covered, I would consider this,which I believe is called a literature review thesis, as an extreme opposite of independent research work thesis).

But what is the "average" level of a MS thesis of a mathematics student? Is it usually closer to a literature review thesis or a research work thesis?

In particular, I would also like to know:

1. 
   

   How much is it valued (if at all) when one applies for PhD?

   
   


2. 
   

   I have heard that its value is more in Europe than America, which if I were to guess I would say, may be due to absence of GRE like criterion there. Is this true?

   
   

Edit: After the wonderful existing answer explaining the case in Germany, I would really like to know the situation in US too. I expect a drastic difference due to the presence of GRE system but would like to know how much importance the thesis has, there.

PS: Please excuse me if one can find answers to some of these questions in already existing questions. I have searched, but couldn't find them. Please provide the links in those cases.

Also, anecdotal details will also be greatly appreciated. Thanks!

Answer

I think this varies a lot. But for Germany your first question can be answers succinctly: In a Master's thesis you should show that you have potential for research.

On the other hand, expectations vary a lot between advisors. But certainly you do not have to prove a new theorem or develop a new theory.

1. How much is it valued (if at all) when one applies for PhD?

I can only answer for the situation where you apply in Germany. The thesis can be a door opener if it is topic closely related to the field where you want to do a PhD. Also a good mark is important. But also in Germany hiring professors will often contact your advisors or request a reference letter and this is much more important.

2. I have heard that its value is more in Europe than America, which if I were to guess I would say, may be due to no GRE like criterion. Is this true?

Not sure on this point since I can't provide a comparison with the US and also I am not sure if the situation is uniform with the EU.