How to write mail to super senior renowned mathematicians? I am 36 year old researcher in field of mathematics. When I get stucked at certain problem I use to write mail to mathematicians expert in the field. Sometime they are of age 80 and above. Most amazing thing is that they respond quickly to every of my mail. Most of the time when I  write mail to young researchers of my age or younger than me I start my mail with word "Dear Sir......." and when I write mail to super senior researchers I start my mail with word "Respected Sir.......".

Is my way of writing  "Respected Sir......." is correct or should use some other words for gesturing greater respect to them?
"Respected Sir" seems off and pretentious, at least in an international setting. It's not commonly used. 

Use "Dear" to be on the save side, and by the way, there's no need to gesture "greater respect" in your greeting. Your appreciation of the researchers previous work and career should be prevalent in the content of your message, in which you demonstrate knowledge of their achievements and capabilities. Attempts at scoring cheap points with cheesy greetings are not the way.