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, 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.
