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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.

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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.

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  • $\begingroup$ Thanks @Alexandre Eremenko!! I always suspected Gauss's Summatorische function is Klein's j invariant. What made me not convinced in it was several numbers and concepts which apear in modern books that i wasn't yet able to find them in Gauss's werke. I mean numbers like 1728 and concepts like the modular discriminant. Can you tell something about Gauss's work in this respect? $\endgroup$ – user2554 Aug 23 '18 at 13:12
  • $\begingroup$ In the book Buhler, Gauss. A biographical study, the author refers to the same page 386 of volume III and explains that hat identification of the "summatorische funktion" mentioned on this page with modular invariant is based on Schlesiger's reconstruction of what Gauss means. This fragment is Gauss' notes not intended for publication, and it is difficult to understand. $\endgroup$ – Alexandre Eremenko Aug 24 '18 at 13:15
  • $\begingroup$ Of course you cannot hope to find modern notation (which is due to Klein and Fricke) in Gauss' work. $\endgroup$ – Alexandre Eremenko Aug 24 '18 at 13:16

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