Well we have been chasing a problem for some time with IE6 where it just seems to stop working on our site which is filled with remote ajax calls. IE7 & Firefox browsers seemed to work just fine but IE6 just didn't seem to work too well. Ill save you to long boring story, but in case you run into this problem, here are some good discussion ideas of how to fix it:
The solution that worked best for us was to add the following line to our apache ssl configuration:
SetEnvIf User-Agent ".*MSIE.*" \ nokeepalive ssl-unclean-shutdown \ downgrade-1.0 force-response-1.0
this forces IE6 to use ssl version 1, which is not as good as whatever it would normally use but their implementation is not broken. I think you lose keep alives out of the deal, but the site actually works now in IE6.
Im amazed about how few web 2.0 ajax enabled companies are using https.