I am Masters student in Mathematics.Currently I am studying modular forms.I am interested in integer partitions, Modular forms. I am working and thinking about problems which are arising during my studies.Besides this I am trying to understand connections of these problems to another field of Maths and physics. Actually I am preparing application for fellowship grant , there is requirement of Research statement.So How can I write down research statement having no strong result? Should false attempts during thinking about problems be included in resarch statement?
It is probably fine to indicate that you have already studied the problems/questions that you are exploring and that you have "narrowed the field" by rejecting certain unfruitful approaches. If you already have "strong results" then the research is behind you, not ahead. Focus more on the importance of the questions you are exploring than the failed attempts to answer them. In mathematics, some problems are inherently difficult. Some are even impossible, but things can be learned and math can advance even for those problems.
It is especially valuable, if true, to point out that some of the attempts to answer the main question have also raised new questions that are fruitful for exploration.