[Paper Review] Divisors of Fourier coefficients of modular forms

Sanoli Gun, M. Ram Murty|arXiv (Cornell University)|Mar 19, 2014

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper investigates the average order of the divisor function applied to Fourier coefficients of normalized Hecke eigenforms of weight $k \geq 2$ for $\Gamma_0(N)$ with integer coefficients. Under the Generalized Riemann Hypothesis (GRH), it establishes that the sum $\sum_{p \leq x,\, a(p) \neq 0} d(a(p))$ grows asymptotically like $x (\log x)^{A}$ for some constant $A$, with explicit bounds derived using Chebotarev density and analytic number theory techniques.

ABSTRACT

Let $d(n)$ denote the number of divisors of $n$. In this paper, we study the average value of $d(a(p))$, where $p$ is a prime and $a(p)$ is the $p$-th Fourier coefficient of a normalized Hecke eigenform of weight $k \ge 2$ for $Γ_0(N)$ having rational integer Fourier coefficients.

Motivation & Objective

• To study the average order of the divisor function $d(a(p))$ over the $p$-th Fourier coefficients of normalized Hecke eigenforms with integer coefficients.
• To establish asymptotic estimates for $\sum_{p \leq x} d(a(p))$ under the assumption of the Generalized Riemann Hypothesis (GRH).
• To extend results on arithmetic functions of modular forms by analyzing the distribution of divisors of Hecke eigenvalues using Galois representations and Chebotarev density.
• To explore whether an asymptotic formula of the form $\sum_{p \leq x} d(a(p)) \sim Bx (\log x)^v$ exists, with $v$ possibly zero.

Proposed method

• Utilizes the representation $\rho_\delta$ attached to the modular form to relate $a(p) \equiv 0 \pmod{\delta}$ to Frobenius conjugacy classes in Galois extensions.
• Applies the Chebotarev density theorem to estimate $\pi(x,\delta)$, the number of primes $p \leq x$ with $a(p) \equiv 0 \pmod{\delta}$, via the density $h(\delta) = |C_\delta| / |G_\delta|$.
• Employs Friedlander and Iwaniec's refinement of van der Corput's method to majorize the divisor function $d(n)$ by short exponential sums.
• Uses Dirichlet series $F(s) = \sum d(n)^c h(n)/n^s$ and $G(s) = \sum d(n)^c / n^s$ with analytic properties to bound partial sums via complex analysis.
• Applies the quasi-GRH to derive weaker but still effective bounds when full GRH is not assumed.
• Combines error terms from GRH and the zero-density estimate for $Z(x)$, the set of primes with $a(p) = 0$, to control the main sum.

Experimental results

Research questions

• RQ1 Can the average value of $d(a(p))$ over primes $p$ be bounded asymptotically under GRH?
• RQ2 Is there an asymptotic formula of the form $\sum_{p \leq x} d(a(p)) \sim Bx (\log x)^v$ for some constants $B$ and $v$?
• RQ3 How does the density $h(\delta)$ of primes $p$ with $a(p) \equiv 0 \pmod{\delta}$ behave, and what role does it play in estimating divisor sums?
• RQ4 To what extent can the full strength of GRH be relaxed while still obtaining power-logarithmic bounds on the sum of divisor functions?

Key findings

• Under GRH, $\sum_{p \leq x,\, a(p) \neq 0} d(a(p)) \ll x (\log x)^{2^c - 1}$ for some constant $c > 0$ depending on the weight $k$.
• The lower bound satisfies $\sum_{p \leq x,\, a(p) \neq 0} d(a(p)) \gg x$, showing the sum grows at least linearly in $x$.
• The error term in the Chebotarev density estimate is $O(\delta^3 x^{1/2} \log(\delta N x))$, which is crucial for controlling the sum over $\delta$.
• The Dirichlet series $F(s) = \sum d(n)^c h(n)/n^s$ is shown to satisfy $\sum_{n \leq z} d(n)^c h(n) \ll (\log z)^{2^c}$, enabling effective partial sum bounds.
• The sum $\sum_{n \leq z} d(n)^c \ll z (\log z)^{2^c}$ is established via $G(s) = \sum d(n)^c / n^s$, which is analytic for $\Re(s) \geq 1$.
• The result remains valid under a quasi-GRH assumption, yielding bounds of the form $x (\log x)^A$ for some $A$, though with weaker explicit constants.