How do researchers ask others for comments and advices for their research? I have self-funded, 8-page long research which is highly suited a famous researcher, in both their academic and personal interests. How should I ask them for comments and advice? I am not asking them to be my advisor or to provide me funding, but a direction to continue my research. A letter of recommendation would be great, of course.

Should the research be written in an attachment, or in the email right ahead? They have to read the research anyway, right? It has a preamble, which conveys the mathematical results in Apple's advertisement style, and has received extremely positive feedback. Should this be placed at the top of the email, even before the salutation? Many presentations do this way, and it does create more impression than the traditional approach. How would other researchers normally do in this situation?



My current approach:


A salutation and a hope that the letter isn't sent when they arn't too busy.
A three-line paragraph just for choosing a proper pronoun. I want to show the respect, but using "professor" multiple times feels distant, which they might not want to be.
A five-line paragraph telling how they fits me (not because they're famous, but because they have niche expertise on the topic), and what my goals are (funding, advice for career path)
A five-line summary of the attachment
A three-line sentence to introduce referees
A thank-you conclusion, re-acknowledge that they're busy but still spending their time reading my letter. Hoping the best for them and their family.


My worry is that they wouldn't open the mail at the first place, or find the summary to be too "astonishing" to be true.
A friend of mine wrote an undergraduate thesis that was groundbreaking original research.  Nobody at his university had the expertise to evaluate it, but one of the faculty suggested he send it to a fairly famous professor at another university.  He did.

That professor read his work and sent him some feedback, including several (correctable) errors he noticed.  He also made sure my friend's application to his program was approved and he was offered their best fellowship.  Eventually my friend attended that doctoral program and the professor he had contacted was his advisor.  He also published the a journal article based on his undergrad thesis in one of the leading journals of his field.

This is an unusual story; most "self-funded" research is not of that quality and many famous researchers don't bother to read that sort of email.  If your work is of that kind of quality, and if you are lucky enough to find a researcher who opens and reads it, this approach can work.  If you believe you're in that situation, definitely do send the manuscript in your initial email (or post it online and provide a link); the likelihood that the researcher will follow up and ask you for a copy, when he/she has no evidence of the quality or relevance of your work, is extremely low.

Always start with a salutation; the last thing you want is to come across as being rude.  I would keep the email short, but long enough to show the researcher that you've done serious research and that it's relevant to his/her interests.  Roughly the length of an abstract, though in your email you can describe things less formally and point out specific connections with the recipient's work.