How should I start Undergraduate Research in Mathematics? Dear Academic SE advisers,

I am a college sophomore in US with double majors in mathematics and microbiology. My algorithmic biology research got me passionate about the number theory and analysis, and I have been pursuing a mathematics major starting on this Spring semester. I have been independently self-studying the number theory textbooks written by Niven/Zuckerman/Montgomery, Apostol, and Ireland/Rosen on this semester. As this semester progressed, I discovered that I am more interested in the pure mathematics than applied aspects (computational biology, cryptography, etc.). I want to pursue a career as analytic number theorist and prove the Collatz conjecture and Erdos-Straus conjecture. 

I have been thinking about doing the number-theory research on my university (research university; huge mathematics department). I have been self-studying the NT by myself and also regularly attending the professional and graduate seminars on number theory but I did not do any pure mathematics research as an undergraduate. Should I visit NT professors in my university and ask them about if I can do undergraduate research under them? If research is not possible (perhaps due to my lacking maturity), should I request of doing independent reading under them and later proceed with the research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask to them or do in my own? 

As for my mathematical background, I have been taking Calculus II (computational) and discrete mathematics. I will be taking calculus III (vector calc.) on Summer, followed by Analysis I, Probability, Theoretical Linear Algebra on Fall 2015. As for my self-studying on this semester, I have been studying NT textbooks (mentioned above), proof methodologies, and basic linear algebra. 

Thank you very much for your time, and I look forward to your advice!

Sincerely,

PK
While it is possible for a highly competent mathematician to dole out a doable problem for an undergraduate to solve over the summer, empirical data has proven otherwise - i.e., it's rather hard for an undergrad to prove anything original in number theory if only given a few weeks during the summer.

For an REU in NT, it might be more realistic to expect to read some interesting topics in number theory or perform some numerical analysis. Take it as a bonus if you obtain any original result.

Number theory is known to be a very difficult topic to get into. Before you decide to commit, take courses in complex analysis and abstract algebra. They are crucial if you want to read more advanced texts in number theory.

Good luck!