What is the best way for an amateur researcher to communicate with an academic? If a  amateur researcher believes that he  has a result (average result,not ground breaking result),   what  should he  do  to communicate  an academic (who is a stranger)?

How should one approach?
There are some general facts to consider here. First, researchers are always busy. This means you only have a few seconds to attract their attention. Second, they're usually knowledgeable in very specific areas of interest. This means, they will be more willing to consider your idea if they know its background already. The "crank" should consider these facts to improve her chances to attract the researcher's attention:


Choose a researcher who has investigated the specific problem addressed by the result before.
The initial e-mail should be brief as well as terse. 3-5 sentences. One sentence should state the problem. One sentence should intelligently summarize the researcher's previous work on the problem. One sentence should summarize the novel idea to solve the problem.
Any more detailed explanations should be attached to the e-mail in a readable and well-organized form, preferably as a PDF file.