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are not similar. [[John Milnor]] observed that the 16-dimensional [[torus|tori]] obtained by dividing '''R'''<sup>16</sup> by these two lattices are consequently examples of [[compact]] [[Riemannian manifold]]s which are [[isospectral]] but not [[isometric]].
The [[Dedekind eta function]] is defined as
:<math>\eta(z) = q^{1/24}\prod_{n=1}^\infty (1-q^n),\ q = e^{2\pi i z}.</math>
Then the [[modular discriminant]] Δ(''z'')=η(''z'')<sup>24</sup> is a modular form of weight 12. A celebrated conjecture of [[Ramanujan]] asserted that the ''q''<sup>''p''</sup> coefficient for any prime ''p'' has absolute value ≤2''p''<sup>11/2</sup>. This was settled by [[Pierre Deligne]] as a result of his work on the [[Weil conjectures]].
The second and third examples give some hint of the connection between modular forms and classical questions in number theory, such as representation of integers by [[quadratic form]]s and the [[Partition function (number theory)|partition function]]. The crucial conceptual link between modular forms and number theory are furnished by the
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