How to approach undergraduate research without official program? I am a second year mathematics and physics student. My primary goal is to enter a post-graduate program in mathematics and hopefully progress further, should I possess the capability. I attend a South African university and have thus far taken the equivalent Calculus I, Precalculus, Numerical Analysis, and Differential Equations I courses. In the new semester I will begin Multivariable Calculus, Linear Algebra, and Analysis before commencing in the subsequent semester with secondary Analysis and Differential Equations courses, alongside Group Theory.

For the past six months or so I have been attempting to make headway on a few questions relating to sums of consecutive prime numbers, typing my (mostly trivial) results in LaTex. I would like to commence undergraduate research of some sort this year, even if it amounts to a professor assigning me problems appropriate for my experience level throughout the year when they can spare the time. Optimally, I would also like to garner the opportunity at some point in the year to discuss some of the questions that I have relating to the prime sums – what topics, textbooks, or papers I could read to progress further, and so on.

Unfortunately, however, my University does not have an official undergraduate research programme (like the URP for example), nor do I believe research-related requests from undergraduate students to be all too common (unlike the University of this poster, from my understanding). What is more is that there are only two Professors in my Department who have research interests in or related to Number Theory (one seems fairly open and is primarily focuses on Partition Theory), and neither of them have been or will be my lecturers going forward.

My main concern is coming across as disrespectful, entitled, or arrogant by reaching out to one of these professors for a potential research opportunity or mentorship.

What would be the generally acceptable way of seeking a research opportunity for a student studying at a University with no formal undergraduate research programme? Should I avoid emails all together, and if not would it be acceptable to mention offhandedly the minor research that I have conducted?

EDIT: I have also found this related post quite useful.
Go and talk to the relevant professors. 

It itself, seeking to make contact is hardly disrespectful -- so long as you are polite of course! Nor is it arrogant -- unless you show up and start teaching the professor their job. And as a university student, you should feel some entitlement to engage with the faculty beyond normal office hours or lectures -- so long as you don't abuse that access.

If you are concerned about appearing brash, send a brief, polite email asking for a meeting to discuss the mathematics you are interested in. If your professors do not want to set aside time to chat with a keen and driven undergraduate student about topics of mutual interest, then more fool them. Bear in mind, always, that professors often have a great deal of pressures on their time. However, I'm willing to bet that chatting about research would be a far more pleasurable activity than many other tasks a professor would have to tackle in the course of his or her day.

Personally, I make time to chat with any undergraduate student about the research I do, and any research they have done on their own accord, or how to further their interests.