I am reading the article "Mathematicians Chase Moonshine’s Shadow" [1], and want to follow up on one of its sources "On the Numbers of Representations of a Number as a Sum of $2r$ Squares, Where $2r$ Does not Exceed Eighteen" by J. W. L. Glaisher [2]. Since I can't get my hand on the original, what would be the best book that summarizes the results of [2]?

[1] https://www.quantamagazine.org/mathematicians-chase-moonshine-string-theory-connections-20150312/

[2] https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms/s2-5.1.479

  • $\begingroup$ I could not find the article sources through that link. Meanwhile, I don't see why Glaisher (1907) should have been listed. You might like the book by Ono, The Web of Modularity. It is the sort of material that would have been in Glaisher $\endgroup$
    – Will Jagy
    Apr 22 '18 at 22:22
  • $\begingroup$ bookstore.ams.org/cbms-102 $\endgroup$
    – Will Jagy
    Apr 22 '18 at 22:23
  • $\begingroup$ It's avaliable on sci-hub $\endgroup$
    – user477805
    Apr 23 '18 at 1:04
  • $\begingroup$ @WillJagy You are right! The article is not directly linked to in the story. The way I got to it was by first looking at "Mock Theta Functions" by Sander Pieter Zwegers. Then Googling on the subject "Mock Theta Functions", and coming across the paper "Theta Functions: The Problem of the Representation of Numbers as Sums of Squares" by Julio C. Andrade. That paper in turn had references to the 1907 paper by Glaisher! :) Crazy twist of events, I know. $\endgroup$ Apr 23 '18 at 13:14
  • $\begingroup$ start with this instead: en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem which gives some modern references. Also en.wikipedia.org/wiki/Sum_of_squares_function $\endgroup$
    – Will Jagy
    Apr 23 '18 at 14:37

After digging through many books, the best I found on the subject are:

  1. Grosswald, E. (1985). Representations of integers as sums of squares. New York: Springer.
  2. Moreno, C. J., & Wagstaff, S. S. (2006). Sums of squares of integers. Boca Raton: Chapman & Hall/CRC.

Both books have references to Glaisher and his work. Both books expand on the subject in detail, using results in the field by other mathematicians (besides Glaisher).


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