This is a slight generalization of a question I made in Math StackExchange, which is still unanswered after a month, so I decided to post it here. I am sorry in advance if it is inappropriate for this site.
Suppose $\Lambda$ is an even lattice. Consider its theta series
$$\theta_{\Lambda}(q) = \sum_{a\in \Lambda} q^{(a,a)/2},$$
where $(\cdot,\cdot)$ denotes the Euclidean inner product.
My question is:
For which $\Lambda$ do we have
$$\theta_{\Lambda}(q) = 1+m\sum_{n>0}\frac{f(n)\: q^n}{1-q^n}$$
where $m$ is nonzero and $f$ is a totally multiplicative arithmetic function?
Examples
I only know of two kinds of lattices with this property:
- Maximal orders in rational division algebras with class number 1, scaled by $\sqrt{2}$:
Dimension 1: The integers, with $m=2$ and $f(n)=\lambda(n)$ is the Liouville function.
Dimension 2: The rings of integers of imaginary quadratic fields of discriminants $D = -3, -4, -7, -8, -11, -19, -43, -67, -163$. Here $m=\frac{2}{L(0,f)}$ and $f(n) = \left(\frac{D}{n}\right)$ is a Kronecker symbol, and $L(0,f)$ is given by $\sum_{n=0}^{|D|} \frac{n}{D} \left(\frac{D}{n}\right)$.
Dimension 4: The maximal orders of totally definite quaternion algebras of discriminants $D = 4, 9, 25, 49, 169$. Here $m=\frac{24}{\sqrt{D}-1}$ and $f(n) = n \left(\frac{D}{n}\right)$.
Dimension 8: The Coxeter order in the rational octonions, with $m=240$ and $f(n)=n^3$.
- The two 16-dimensional lattices of heterotic string theory, $E_8\times E_8$ and $D_{16}^+$. Both lattices have the same theta series, with $m=480$ and $f(n)=n^7$.
These include in particular all the root lattices I mentioned in the original Math.SE post.
Attempt
(Feel free to skip this part)
I do not know much about modular forms so this may contain mistakes. Theorem 4 in these notes implies that in even dimension there is a level $N$ and a character $\chi$ taking values in $\{-1,0,1\}$ for which $\theta_{\Lambda}$ is a modular form of weight $k = (\mathrm{dim}\: \Lambda) /2$. The requested property, in turn, implies that the Epstein zeta function of the lattice has an Euler product
$$\zeta_{\Lambda} (s) \propto \prod_p \frac{1}{1-(1+f(p))p^{-s}+f(p)p^{-2s}} = \zeta(s) \prod_p \frac{1}{1-f(p)p^{-s}},$$
which in even dimension means that $\theta_{\Lambda}$ is a Hecke eigenform (noncuspidal, given the leading coefficient 1); we thus see that it must be an Eisenstein series of weight $k$, level $N$ and character $\chi$, by the decomposition of the space of modular forms into Eisenstein + cuspidal subspaces.
This Eisenstein series has the Fourier expansion $E_{k,\chi}(q) = 1- (2k/B_{k,\chi}) \sum (\cdots)$ where $B_{k,\chi}$ is a generalized Bernoulli number and the $(\cdots)$ part has integral coefficients. So one possible course of action would be to find those generalized Bernoulli numbers for which $2k/B_{k,\chi} = -m$ is a negative even integer (since in $\Lambda$ there must be an even number of vectors of norm 2), and check case by case whether the associated Eisenstein series is the theta series of a lattice.
If this approach is correct, we can then use Tables 1-3 in this paper, which shows that the only such cases with $\mathrm{dim}\: \Lambda \ge 4$ are the ones given in the Examples section, together with a certain Eisenstein series of weight 2 and level 42, which does not seem to correspond to a lattice.
On the other hand, I don't understand what happens in the odd-dimensional case (apart from dimension 1 which is trivial), where the modular forms involved are of half-integral weight. It seems that the concept of Hecke eigenform is defined a bit differently, so the above approach may not work here. I found this answer which says that zeta functions associated to modular forms of half-integer weight generally lack Euler products. Here are also some possibly relevant questions (1, 2) dealing with special cases. Another particular case, namely products of powers of the Jacobi theta function and Dedekind eta function, was treated in this paper by Hecke himself.