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Except for Eisenstein series having the divisor functions as their Fourier coefficients, is there any other full integral weight modular form (of some level, preferably full) having arithmetic functions as their Fourier coefficients.

More to the point, my question is, apart from the relations you obtain between $\sigma_3, \sigma_5$ and $\sigma_7$ are there any applications of full integral weight modular forms (preferably cusp forms) to elementary number theory.

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Formulas for sums of squares. Groasswald's book has a chapter dedicated to the connection. –  Dror Speiser Jul 12 '11 at 16:39
I don't recall all the details, or know a good reference off the top of my head, so I will leave a comment rather than an answer: Suppose $a_1 x_1^2 + \cdots + a_k x_k^2$ is a quadratic form with all $a_i > 0$. Let $r(n)$ be the number of representations of $n$ by this quadratic form. Then $\sum_n r(n)$ q^n is a modular form of weight $k/2$, although not usually of full level. You can use this to conclude facts about the $r(n)$. –  Frank Thorne Jul 12 '11 at 16:42
You can see Ono, The Web of Modularity for all sorts of interesting examples where a variety of arithmetic functions show up as coefficients of modular forms. –  Frank Thorne Jul 12 '11 at 16:43
Counting sums of squares is a good example, though the forms that arise aren't quite for the full modular group. Probably the closest that this direction comes to "elementary" would be representations by the quadratic form associated to the even unimodular lattice $D_n^+$ for $n \equiv 0 \bmod 8$ with $n \geq 24$. If you try to extend the $\sigma$ convolutions past 3,5,7 you run into discrepancies coming from coefficients of cuspforms. But there are linear combinations that still work, starting with $441 E_4^3 + 250 E_6^2 = 691 E_{12}$. –  Noam D. Elkies Jul 12 '11 at 16:55
Hi Mehmet! You might like the chapters concerning quadratic forms in Iwaniec's "classical topics" book. –  David Hansen Jul 13 '11 at 2:22
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4 Answers

up vote 10 down vote accepted

Perhaps this is "out of bounds" given the phrasing of the question, but those Eisenstein series you mention don't just have divisor sums as coefficients - the constant term is a special value of the Riemann zeta function.

This implies all sorts of neat stuff. The various relations between divisor sums that you mention come with relations between zeta values. These give very nice congruences, in particular.

You can take this to reasoning pretty far to deduce things like $p$-adic interpolation of zeta values a la Kubota-Leopoldt from the much simpler interpolation properties of the divisor sum functions. This was done by Serre in the 1973 paper ("Formes modulaires et fontiones zeta $p$-adiques") that gave birth to the theory of $p$-adic modular forms.

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Personally, I regard the Dirichlet coefficients of any automorphic $L$-function arithmetic. A nice application of the Fourier coefficients of full level Maass cusp forms is Motohashi's improvement for the error term in the binary additive divisor problem and the asymptotic formula for the fourth moment of the Riemann zeta function. See in particular his papers here and Theorem 5.2 in his book.

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Another little thing: the simplest Siegel-Weil formulas, equating holomorphic Eisenstein series and linear combinations of theta series attached to positive-definite quadratic forms, can be arranged to be about level-one or small-level things.

Edit: and add Klingen's proof of rationality properties of special values of zeta functions of totally real number fields.

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Nobody seems to have mentioned "the master" of this subject, and his use of (classical) Eisenstein series to prove things like $p(5n+4) \equiv 0 \ ({\rm mod} \ 5)$ (here $p(m)$ is of course the usual partition function).

Here's his proof (prepared by Hardy), published in Math.Z (1921). B.Berndt published another a bit shorter proof , which employs famous Ramanujan's differential equations. BTW, make sure you are familiar with the Ramanujan "J-series" before you jump to (say) formula (2.2) in Berndt's paper :-)

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