Representation of Ramanujan's tau function by twisted divisor functions

1. Problem Statement

The paper addresses the classical problem of finding explicit formulas for Ramanujan's tau function, $\tau(n)$, which is defined as the Fourier coefficients of the discriminant modular form $\Delta(z)$ of weight 12 and level 1:
$$\Delta(z) = q \prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty \tau(n)q^n, \quad q = e^{2\pi i z}.$$
While various identities expressing $\tau(n)$ as convolution sums of standard divisor functions (e.g., $\sigma_k(n)$) exist (notably by van der Pol, Lehmer, and others), this paper seeks to generalize these results. Specifically, it aims to represent $\tau(n)$ using convolution sums of twisted divisor functions associated with higher-level modular forms. The goal is to establish an infinite family of such identities parameterized by an odd square-free integer $D$ and Dirichlet characters.

2. Methodology

The author employs a constructive approach involving the theory of modular forms, specifically utilizing Rankin-Cohen brackets and trace maps. The methodology proceeds in four main stages:

1. Construction of Level 1 Cusp Forms from Higher Levels:
   The core strategy involves constructing a cusp form of level 1 (weight 12) by taking the trace of Rankin-Cohen brackets of Eisenstein series defined at higher levels ($D$).

   • Let $G_{k, \chi}(z)$ be Eisenstein series of weight $k$ and level $D$ with character $\chi$.
   • The author defines a family of forms $F_{D, \ell, k, e}(z; \chi)$ by taking the trace map $\text{Tr}_1^D$ of the $e$-th Rankin-Cohen bracket of two such Eisenstein series:
     $$F_{D, \ell, k, e}(z; \chi) = \text{Tr}_1^D \left( [G_{\ell, \chi}(z), G_{k, \chi}(z)]_e \right).$$
   • The Rankin-Cohen bracket $[f, g]_e$ is a differential operator that produces a cusp form from two modular forms.

2. Proof of Non-Vanishing:
   A critical hurdle is proving that the constructed form $F_{D, \ell, k, e}$ is not identically zero (which is necessary to normalize it and compare it to $\Delta(z)$).

   • The author uses the Rankin-Selberg method to compute the Petersson inner product $\langle f, F_{D, \ell, k, e} \rangle_1$ for a normalized Hecke eigenform $f$.
   • This inner product is shown to be proportional to products of twisted L-values (specifically $L(f, s)$ and $L(f, \chi, s)$).
   • By invoking known non-vanishing results for these L-values (e.g., Kohnen-Zagier), the author proves that $F_{D, \ell, k, e}$ is non-zero in the relevant cases.

3. Uniqueness Argument:
   Since the constructed forms $F_{D, \ell, k, e}$ are cusp forms of weight 12 and level 1, and the space $S_{12}(1)$ is 1-dimensional (spanned uniquely by $\Delta(z)$), any non-zero normalized form in this space must be equal to $\Delta(z)$.

   • Therefore, $\Delta(z) = \frac{F_{D, \ell, k, e}(z; \chi)}{a(1)}$, where $a(1)$ is the first Fourier coefficient.

4. Explicit Computation of Coefficients:
   The author explicitly computes the Fourier coefficients of the trace of the Rankin-Cohen brackets. This involves:

   • Decomposing the level $D$ into factors $D = D_1 D_2$.
   • Utilizing the transformation properties of Eisenstein series under the slash operator.
   • Deriving a closed-form expression for the coefficients involving sums of twisted divisor functions $\sigma_{k-1, \chi_1, \chi_2}(n)$.

3. Key Contributions and Results

A. Definition of Twisted Divisor Functions

The paper defines twisted divisor functions $\sigma_{l-1, \phi, \psi}(n)$ associated with Dirichlet characters $\phi$ and $\psi$ modulo $D_1$ and $D_2$ (where $D=D_1 D_2$):
$$\sigma_{l-1, \phi, \psi}(n) = \sum_{d_1 d_2 = n} \phi(d_1)\psi(d_2)d_1^{l-1}.$$

B. Main Theorem (Theorem 1.1)

The paper establishes an infinite family of identities representing $\tau(n)$ in terms of these twisted functions. Let $\tilde{a}_{D, \ell, k, e}(n; \chi)$ be the normalized coefficient of the constructed form. The theorem states:

1. Case $\chi(-1) = -1$ (Odd character):
   $$\tau(n) = \tilde{a}_{D, 3, 5, 2}(n; \chi) \quad \text{and} \quad \tau(n) = \tilde{a}_{D, 3, 7, 1}(n; \chi).$$
2. Case $\chi(-1) = 1$ (Even character):
   $$\tau(n) = \tilde{a}_{D, 4, 6, 1}(n; \chi).$$
3. Special Case ($D=1$ or $D \equiv 5 \pmod 8$):
   $$\tau(n) = \tilde{a}_{D, 4, 2}(n).$$

These identities generalize classical results. For instance, when $D=1$, the formulas reduce to known identities involving standard divisor functions $\sigma_k(n)$ (e.g., the van der Pol type identities).

C. Numerical Examples and Applications

• Numerical Verification: The author provides explicit numerical tables for $D=7$ and various characters, verifying that the normalized coefficients match $\tau(n)$ exactly.
• Evaluation of L-functions: The paper demonstrates an application where these identities can be used to evaluate Dirichlet L-values (specifically $L(-3, \chi_p)$) without relying on Bernoulli polynomials, offering an alternative formula derived from the convolution sums.

4. Significance

• Generalization: The work significantly extends the scope of known identities for $\tau(n)$ from level 1 to arbitrary odd square-free levels $D$, introducing the concept of "twisted" divisor functions into the representation of the tau function.
• Methodological Innovation: It provides a robust framework for generating new identities by combining Rankin-Cohen brackets with trace maps, bypassing the need for ad-hoc construction.
• Non-Vanishing Proof: The rigorous proof of non-vanishing via the Rankin-Selberg method and L-value theory resolves a potential gap in constructing cusp forms from higher levels, ensuring the validity of the normalization.
• Conjectures: The paper concludes with two conjectures:
  1. The non-vanishing of the first Fourier coefficient $a_{D, \ell, k, e}(1; \chi)$ for sufficiently large weights.
  2. The assertion that the constructed forms are not Hecke eigenforms when the dimension of the cusp space is greater than 1.

In summary, Tianyu Ni successfully constructs a bridge between higher-level modular forms and the classical Ramanujan tau function, providing a rich family of new arithmetic identities and a deeper understanding of the structure of $\Delta(z)$ through the lens of twisted divisor sums.