Relation between arithmetic functions and modular forms

Question

Generating functions of many multiplicative arithmetic functions of longstanding interest (e.g., sum-of-divisors function, number-of-partitions function) turn out to be Fourier expansions of modular forms.

Is this true of all multiplicative arithmetic functions? If $\gamma : \mathbb{N} \rightarrow \mathbb{C}$ is a multiplicative arithmetic function (i.e., $\gamma(n m) = \gamma (n) \gamma (m)$ whenever $n$ and $m$ are relatively prime), is the formal sum $\sum_{n=1}^\infty \gamma(n) q^n$ (related to) Fourier expansion of some modular/automorphic form?

A different way of putting the question is: is there a direct relation between the multiplicative property of arithmetic functions and the symmetry of modular/automorphic forms under the action of associated modular/reductive group, or is it the case that some arithmetic functions just happen to have modular forms as generating functions without there being any larger conceptual relationship between the two?

Answer 1

The reason for multiplicativity of some modular forms is due to their behavior under Hecke operators. The Wikipedia article states:

If a (non-zero) cusp form $f$ is a simultaneous eigenform of all Hecke operators $T_m$ with eigenvalues $\lambda_m$ then $a_m = \lambda_m a_1$ and $a_1\ne 0$. Hecke eigenforms are normalized so that $a_1=1,$ then $$ T_m f=a_m f,\quad a_m a_n = \sum_{r>0,r|(m,n)} r^{k-1}a_{nm/r^2}, \; m,n\ge 1. $$

In such a case, if $(n,m)=1$ then $a_ma_n=a_{nm}$ which implies that $a_n$ is multiplicative. Futhermore, the function $a_n$ satisifies the stronger property as given in the equation when $(n,m)>1.$

The process does not go the other way. Any multiplicative arithmetic function is completely determined by its values at prime powers and only in very special cases will this be the Fourier coefficients of a modular form. However, these special cases are very interesting. The OEIS contains several examples of these cases including a few infinite families. Other special case come from eta-quotients. Some 74 of them are listed in Yves Martin, "Multiplicative eta-quotients", Trans. Amer. Math. Soc. (1996).

Important special cases arise from elliptic curves. For every elliptic curve over the rationals there is a unique weight 2 multiplicative modular form with corresponding L-series. This is part of the modularity theorem which was key to proving Fermat's last theorem.