Representation of Ramanujan's tau function by twisted divisor functions

Abstract: We present an infinite family of identities that represent Ramanujan's tau function in terms of convolution sums of twisted divisor functions. Our method involves explicitly constructing non-vanishing level $1$ cusp forms from modular forms of higher levels.

• The paper introduces an infinite family of explicit identities expressing Ramanujan’s tau function via convolution sums of twisted divisor functions.
• It employs Rankin-Cohen brackets, trace maps, and higher-level modular forms to construct level 1 cusp forms with explicit Fourier coefficients.
• The methods yield numerical confirmations and non-vanishing results that link tau function representations with twisted L-values, advancing computational number theory.

Representation of Ramanujan's Tau Function by Twisted Divisor Functions

Introduction and Motivation

The paper "Representation of Ramanujan's tau function by twisted divisor functions" (2604.13365) presents a new infinite family of explicit identities expressing Ramanujan's tau function $\tau(n)$, the Fourier coefficients of the unique normalized cusp form $\Delta(z)$ of weight 12 and level 1, in terms of convolution sums of twisted divisor functions. Previous works, including classical identities by van der Pol and later generalizations, have established formulas for $\tau(n)$ via convolution sums involving standard divisor functions. The novelty of the present work lies in employing convolution sums of twisted divisor functions, which arise from higher-level modular forms and Dirichlet characters, enabling a parametrized generalization beyond the classical level 1 context.

The theoretical underpinning combines explicit constructions of level 1 cusp forms from modular forms of higher levels, utilizing Rankin-Cohen brackets and trace operations, and yields formulas with parameters reflecting the level and the twisting character.

Main Results and Formulas

Let $D$ be an odd square-free integer, $\chi$ a primitive Dirichlet character modulo $D$, and $\sigma_{l-1,\phi,\psi}(n)$ denote the twisted divisor function associated to Dirichlet characters $\phi$ and $\psi$. The crux of the paper is the construction of quantities $a_{D,\ell,k,e}(n;\chi)$, which are explicit convolution sums of twisted divisor functions. These are normalized by their first coefficient to yield $\Delta(z)$0.

The central theorem asserts:

• If $\Delta(z)$1, then

$\Delta(z)$2

• If $\Delta(z)$3, then

$\Delta(z)$4

• If $\Delta(z)$5 or $\Delta(z)$6 is a prime congruent to $\Delta(z)$7 mod $\Delta(z)$8, then

$\Delta(z)$9

These identities generalize the classical van der Pol-type formulas and extend them to encompass cases involving twisted divisor functions and nontrivial Dirichlet characters. In particular, the formulas for $\tau(n)$0 are highly explicit, involving sums over partitions of $\tau(n)$1 and weighted sums of twisted divisor functions, as detailed in Proposition \ref{prop:fouriercoexplicit}.

Strong numerical confirmation is provided: for $\tau(n)$2 and nontrivial odd primitive Dirichlet characters modulo 7, the computed values $\tau(n)$3, after normalization, reproduce the first ten values of $\tau(n)$4 exactly.

Construction of Level 1 Cusp Forms via Higher-Level Modular Forms

The technical heart of the paper lies in constructing new cusp forms of level 1 from modular forms of higher levels using Rankin-Cohen brackets, Eisenstein series, and trace maps. Given two Eisenstein series $\tau(n)$5 and $\tau(n)$6 of respective weights $\tau(n)$7 and $\tau(n)$8 and level $\tau(n)$9, their $D$0-th Rankin-Cohen bracket $D$1 yields a cusp form of weight $D$2 and level $D$3. Applying the trace map from level $D$4 to level 1 produces a level 1 cusp form.

Explicit computations of Fourier coefficients are furnished, using detailed expansion techniques and the $D$5-operator, yielding convolution sums involving twisted divisor functions. This explicitness enables the identities for $D$6 as stated, and pushes forward the landscape of explicit formulas in the theory of modular forms.

Non-vanishing and Analytic Implications

Establishing that these constructions yield nontrivial forms (not identically zero) is nontrivial. The author invokes the Rankin-Selberg method, relating the Petersson inner product of the constructed forms to twisted $D$7-values. In particular, non-vanishing results hinge on non-vanishing of twisted $D$8-values at certain points, a deep theme in analytic number theory.

General conditions for non-vanishing are proved for the constructed forms, and the analytic argument circumvents the lack of combinatorial or algebraic guarantees for non-vanishing of specific Fourier coefficients.

A notable application is the derivation of explicit formulas for Dirichlet $D$9-values at negative integers, illustrating how these identities provide alternatives to classical Bernoulli polynomial formulas and serve computational purposes in analytic number theory.

Implications and Prospects

The results yield substantial implications for the explicit theory of modular forms:

• Parametric Families of Identities: The infinite parameter family produced, involving $\chi$0, $\chi$1, $\chi$2, $\chi$3, and $\chi$4, expands the toolkit for expressing $\chi$5, and extends potential for further explicit formulas for other modular forms.
• Connections with Twisted $\chi$6-values: The method connects the arithmetic of $\chi$7 to twisted $\chi$8-values, providing new routes for analytic exploration and computational evaluation.
• Potential for Computational Number Theory: The explicit formulas invite practical computation, and the author provides concrete numerical evidence, as well as code for computational verification.
• Theoretical Directions: The construction and trace technique can be applied to modular forms of other weights, and conjectures are posed regarding the general non-vanishing of the Fourier coefficients for high weight, as well as the characterization of when the constructed forms are or are not Hecke eigenforms.

Future developments may include generalization to higher-weight cusp forms, refinement of non-vanishing criteria, explicit formulas for periods and $\chi$9-values in greater generality, and computational algorithms rooted in these identities. The connection with period polynomials, Rankin-Cohen brackets, and the combinatorics of modular forms suggest that these techniques may have further reach in the explicit arithmetic of automorphic forms.

Conclusion

This paper establishes an infinite family of explicit identities representing Ramanujan's tau function $D$0 via convolution sums of twisted divisor functions, exploiting constructions from higher-level modular forms. The analytical approach via Rankin-Selberg and exposition of explicit Fourier coefficients contribute significant formulas both theoretically and computationally. The interplay between modular forms, Dirichlet characters, Rankin-Cohen brackets, and twisted $D$1-values enriches the explicit arithmetic theory of modular forms and heralds avenues for further formal and computational explorations in analytic number theory (2604.13365).