2
$\begingroup$

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 $\chi$.

$\endgroup$
0

2 Answers 2

6
$\begingroup$

Since you allow $r \in \mathbf Q$, what is the point of writing $12$ in $e^{i\pi nr/12}$? It might as well be written as $e^{i \pi nr}$ by replacing $r$ with $12r$.

In any case, your question has a negative answer when $r = 0$: the series $\sum_{n \geq 1} \psi(n)/n$ converges. The series $\sum_{n \geq 1} \psi(n)/n^s$, which converges absolutely when ${\rm Re}(s) > 3/2$, is $L(E,s)$ where $E$ is the elliptic curve here, and it turns out that the Dirichlet series for each elliptic curve over $\mathbf Q$ converges when ${\rm Re}(s) > 5/6$, so in particular it converges at $s = 1$. A reference where that convergence is proved is in my answer here.

$\endgroup$
1
  • $\begingroup$ Great thanks ! I tried deducing the convergence for all rationals by elementary methods (Tauberian theorems, etc), but was unsuccessful : all these were "purely analytic methods" which also applied to irrational values of $r$, so they had to fail. Have similar results been showed for the L-series twisted by characters $\bmod{N}$ ? If so that would imply the convergence of certain linear combinations of my sums at various rationals (which still does not deal with all the cases...). $\endgroup$ Feb 24 at 21:14
0
$\begingroup$

Let me sum up my current state of knowledge.

First, here are some known facts about the coefficients $\psi(n)$ taken from the 1975 article by Ligozat called Courbes modulaires de genre 1.

  • Théorème B (2.2.3): The Dirichlet $L$-series $L(\psi,s)=\sum_1^\infty \frac{1}{n^s}\psi(n)$ attached the weight $2$ cusp form $\eta^4(6\tau)$ for the group $\Gamma_0(36)$, coincides with the $L$-series attached to the elliptic modular curve $X_0(36)$ (with Weierstrass form $y^2=x^3-1$).

  • Théorème C (2.3.2): The function $\Xi(s)=\left(\tfrac{6}{2\pi}\right)^s \Gamma(s) L(\psi,s)$ admits an analytic continuation to an entire function which satisfies the functional equation $\Xi(s)=\Xi(2-s)$ and is bounded in the vertical strip (i think the strip $0\le \Re(s) \le 2$).

We may thus apply Section 2.2 of the 1993 article by Murty called Modular elliptic curves (mentioned by @KConrad in his comment) to deduce that for all $\epsilon>0$ we have $\lvert \sum_1^N \psi(n)\rvert = o(N^{5/6+\epsilon})$, hence the $L$-series converges where $\Re(s)\ge 5/6$. This proves convergence at $r=0$.

I expect that the convergence at other rational $r\in \mathbb{Q}$ could follow from a similar argument. First prove a functional equation for series $\sum_1^\infty \frac{1}{n^s}\psi(n)e^{i2\pi nr}$, and its analytic continuation to an entire function in a vertical strip containing $1$. Then deduce a bound of the form $\lvert \sum_1^N \psi(n)e^{i2\pi nr}\rvert = o(N^{1-\epsilon})$ whence convergence of the series at $s=1$.

The closest references we found are

  • the 1993 by article by Yogananda called Transformation formula for exponential sums involving Fourier coefficients of modular forms.
  • the 2023 arxiv post by Liu called On sums of Fourier coefficients of cusp forms twisted with additive characters, which deals with cusp forms for the full modular group $\Gamma(1)$.

These references come very close to answering my question...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.