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(Answer to Question 2) The easiest application of modular forms to understand is in classic analytic number theory. Modular forms often act as generating functions for several interesting arithmetic functions. I will give three examples: 1) the eta function almost generates for the number of integer partitions of $n$, 2) Powers of theta functions generate the number of ways a number can be written as a sum of squares, 3)Eisenstein series generate weighted divisor functions.