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Another occurrence of modular forms is in remainder terms of many asymptotic formulas in number theory. Eisenstein series have "large" coefficients, which are explicitly known, while cusp forms have "small" coefficients, which are usually less explicit. By "explicit" in this context we mean that the coefficients are given by simple arithmetic functions, for example divisor functions. If the spaces of modular forms that we consider are finite dimensional and if by any chance they are spanned by Eisenstein series, we can obtain explicit formulas for the coefficients of modular forms, with no error term. On the other hand, if they are not spanned by Eisenstein series, we can still write explicit formulas, with a "small" error term corresponding to the contribution of cusp forms.

A typical example of this is to find formulas for $r_k(n)$, the number of representations of a positive integer $n$ as a sum of $k$ squares. This is in fact the coefficient of $e^{\pi i z}$ in θ$^k$. The space of modular form to which $\theta^k$ belongs is known to be spanned by Eisenstein series for $k\le8$, so we have explicit formulas for $r_k(n)$ in these cases and otherwise we have approximate explicit formulas with a small error term. In fact, in the special case $k=10$, the cusp form which occurs is a CM form, so it has the special property that its coefficients can also easily be computed, so there does also exist an explicit formula for $r_{10}(n)$.

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  • Review status: beta
  • Last edited by Andreea Mocanu on 2016-04-01 12:36:46
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