A basic problem in analytic number theory is to study averages of arithmetic functions. We shall be interested mainly in the class of multiplicative functions, which includes the function counting the number of divisors of a positive integer, also known as the divisor function. One of the first tools to investigate the average order of the divisor function was introduced by Dirichlet, and is referred to as the Dirichlet Hyperbola Method. While this method is certainly useful, it is usually insufficient by itself to deal with a finer question, namely when the sum is taken over sparse sequences. To illustrate this phenomenon, we will first survey some classical results concerning the divisor function over polynomial values. Then we will consider a variant of this problem in the context of modular forms, which are periodic complex functions that satisfy many internal symmetries. More precisely, we will determine the order of magnitude for the divisor sum over the Fourier coefficients of a modular form.