## Definition

Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$,

which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity:
$$\eta^4(6\tau) = q \prod_1^\infty (1-q^{6n})^4 = \sum_1^\infty \psi(n) q^{n/4}.$$

One may also express $\psi(n)$ as the sum of $\chi(abcd)$ over $\{(a,b,c,d)\in \mathbb{N}^4 \mid a^2+b^2+c^2+d^2=n\}$ where $\chi$ is the unique primitive character $\bmod{12}$ (that is the Jacobi symbol $\bmod{12}$, equal to $\chi(12n\pm 1)=1$, $\chi(12n\pm 5)=-1$ and $\chi(n)=0$ if $2\mid n$ or $3\mid n$).

I will recall the most important facts about this sequence as we go.

## Question

I would like to show that for all rational $r \in \mathbb{Q}$ the following series diverges: $$\sum_{n=1}^\infty \tfrac{\psi(n)}{n}\cdot \exp\left(i\pi nr\right).$$

## Elements in favor of divergence

I can show that it is not absolutely convergent. Indeed, we have $\psi(p)\ne 0$ for primes $p\equiv 1\bmod{6}$, and Dirichlet's density theorem implies that the sum $\sum \tfrac{1}{p}$ over primes $p\equiv 1\bmod{6}$ diverges.

I also have a geometric proof (way too long to explain) that the same series replacing $r$ by a real irrational number is always divergent, and that there is a dense subset of the real numbers for which its modulus diverges to $+\infty$.

I computed the partial series as far as my computer could go, and i observe that they start to converge in some region, but then they oscillate in that small region. Maybe there is some very slow convergence which makes it invisible to the experiments, but so far this oscillation has an amplitude which seems to be neither monotonous nor decreasing.

## Elements in favor of convergence

The function $\eta^4(6\tau)$ is a Hecke modular form for the group $\Gamma_0(36)$, which is associated to the elliptic modular curve $X_0(36)$. In particular it satisfies the Ramanajan-Petersson conjecture, namely for $p$ prime we have $\lvert \psi(p)\rvert \le 2 \sqrt{p}$.

The the sequence $\psi(n)$ is lacunary in the sense that $Card\{n\in \mathbb{N}\mid \psi(n)\ne 0, \, n<N\} = o(N)$.

## Final remark

The conditional convergence of the sums would imply that there is a strong de-correlational between $\psi(n)$ and finite rotations of the circle (in the sense of "M"obius disjointness").

This may be equivalent to some kind of "Dirichlet equidistribution theorem" for the arithmetic progressions represented by a the quadratic form $a^2+b^2+c^2+d^2$ or by the divisor function $\sigma(n)=\sum_{d\mid n} d$, twisted by $\chi$.