The short of it… NO!
The 2 algorithms are separate and they the organic algorithm ignores the adwords algorithm.
I have come across the question a couple of times and one person that was certain that there was a direct correlation.
In this instance it was a mis-configuration of their analytics, which was not flagging their adwords traffic correctly so they presumed it was organic.
With all of this in mind and although there is no benefit on organic ranking (directly from the using adwords), I have seen an impact on organic rankings (when used correctly) as a result of paid ads.
This has happened when trying to increase the quality score on an ad for adwords, which indirectly impacted the keyword density of the page that was ranking organically.
The moral of the story here… Make sure you use landing pages in your campaign and keep your adwords and organic separate.
Also, if you use you adwords data (CTR and Bounce Rate) in correlation with your Organic struture, you can improve your overall traffic.
Sources, Sources, Sources…
You’ll notice I’m only grabbing sources that are direct from Google as these are monitored and audited by Google.
The people that have responded have earned a higher member level (which is earned from Google based on accuracy, number of posts and knowledge)
Also, this is a youtube video from Matt Cutts (from Google) where he states there is no correlation – http://www.youtube.com/watch?v=Dvdg17aS5iI