How can I model a "multiplier effect" in time series data?

#1
I currently have data corresponding to how often a certain set of songs were downloaded. Each song has a release date, and then the number of downloads per day going forward to today. It would look like:

Code:
                                   Song1    Cumulative Sum of Song1
    Release Date   5/7/2013         20       20
                   5/8/2013         40       60
                   5/9/2013         55       115
                   .................................



                                    Song2    Cumulative Sum of Song2
    Release Date   7/9/2013          13       13
                   7/10/2013         44       57
                   7/11/2013         92       149
                   .................................

I have about 500 songs total with data just like this.

I am curious to see if I can describe some sort of a "multiplier" effect for each song. What I mean by this is if the customers would be somehow influenced by how many downloads there currently are. Hence, if it is a positive influence, we should see more downloads each day as the cumulative sum gets larger.

Would there be a model seeing what "multiplier" effect the cumulative sum column has on the song downloads for a next day? I tried to think about something like a logit model, but it works only for a binary dependent variable and a standard linear regression seems too easy. Any ideas or advice would be GREATLY appreciated! Thanks!