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Background of question ( Background is somewhat related to this question Asking about how to minimize the effect of a not so good thesis due to abandoning by supervisor without even telling in Phd Application Form but question is entirely different)

I am a masters 2nd year student( 4 th semester)in mathematics .During my masters degree I self studied 3 books of number theory ( 1 elementary,2 analytic)because I was really interested in learning it and no faculty in my Institute is specialized in it( most people in my Institute are specialized in Applied Mathematics) . I am in an 3rd world country.

I want to do phd in number theory in Europe. So, for my Masters thesis I wrote to one of professors in a top Institute in my country and she said yes.I visited her Institute in January,2020 for 8 days but she abandoned me and didn't even bothered to tell as she didn't replied my e - mails and despite the fact that she gave me a research paper in January to study and after completing it I e-mail her on 20 Feb asking her to tell me which paper to study now/ problem to work and when should I visit her Institute again . I e mailed her on 29 Feb, 7 March, 25 March, 4April.( in between I read 2 papers to know a bit more about the work in the area).

Also I am 100% sure that she is not suffering from covid. So, I talked to a professor in my institute and he took absolutely no interest in helping me and send prepare a report from the paper what you have studied and hung up.

I have read 4 papers in analytic number theory till now( all are related to 1st paper given by professor) . And i am thinking of reading another research paper in analytic number theory but it's related to a different problem if time permits. But i have not done any original work like writing a research paper or proving a theorem.

My questions are 1.what kind of work is expected in a Masters thesis by the Admission committees of European Institutes ?

2.Is is necessary to have a publication or original work like proving some results for applying to European Institutes .

Please guide. I have no other sources of help. I wanted to work really nicely for the thesis but things changed completely due to negligence of that professor. Even the professor of my Institute didn't bothered.

I am in desperate need of guidence.

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    You're asking a lot in one post. Please try to break your post down into only asking one question. Also, your questions are mostly specific to you, the universities you're applying to, and you're specific situation, which makes them off topic to this site. – Richard Erickson Apr 22 at 19:58
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    @Richard Erickson I am editing it. But how it's off topic ? Others can also face similar type of situation!! – user795826 Apr 23 at 5:48
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From a Finnish perspective (the department was focused on analysis, not number theory):

  1. No, it is not expected for you to have original publications from your master studies, but if you have them, great! I am not sure how a publication in a predatory or very low quality journal would be taken; in the spirit of being careful, I suggest not publishing unless you are certain it is a decent result and a decent journal. Or maybe only publish a preprint.

  2. A good master's thesis need not include original research, but it might. It shows a clear grasp of mathematical thinking and good and non-trivial proofs. An excellent thesis has original work, or does the same as a good one in a very demanding subject.

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