So I am writing a research paper on the properties of the Dirichlet function (the function with 1 if x is rational and 0 if x is irrational), and I wanted to include some historical background on how Dirichlet came up with this function. However, I have found next to nothing about how he came up with this function. All I have found are what the function is and how you prove it, and other applications. But I want to include something on the historical developments of the Dirichlet function for my research paper.
Do anybody know of the historical background to the Dirichlet function? And if so, what resource are you getting this information from?