# What is the modern interpretation of Gauss's “Summatorische Function”?

In Buhler's biography of Gauss (Gauss: A Biographical Study), at the chapter on modular forms and hypergeometric series, he mentions a function that Gauss called "Summatorische Function", which he used implicitly without giving an explicit definition to it. According to Buhler, "it is in fact the absolute invariant of the modular group of all linear substitutions $t' = \frac {{\alpha t - i\beta}}{{\delta + i\gamma t}}$ where $\alpha\delta - \beta\gamma = 1$ and $\alpha,\beta,\gamma, \delta$ are integers".

So, my first question is how is this function interpreted from a modern point of view?

My other questions are also relevant to this post. In Wolfram MathWorld article on "j-Function" appears the fact that "Gauss was apparently aware of the j-function before 1800". Also, the other name of the j-function ("Klein's absolute invariant") suggests there might be a connection with the Summatorische function that Buhler mentions. So, where in Gauss's nachlass is the relevant work on the j function? and is it connected with the Summatorische function?

Since modular forms are a very advanced mathematical topic (i think mathematicians begin to learn about it only in their second degree), obviously i wasn't able to understand the answer to my question by myself.

According to Klein (Lectures on history of mathematics in 19 century), he absolute invariant $J$ was introduced by Gauss in his manuscript "On summatory function", which is reproduced on p. 386 of volume III of Gauss collected works.