[Paper Review] On the value-distribution of symmetric power L-functions

Kohji Matsumoto, Yumiko Umegaki|arXiv (Cornell University)|Aug 17, 2018

Analytic Number Theory Research26 references3 citations

TL;DR

This paper establishes limit theorems for the value-distribution of symmetric power L-functions, extending the Bohr-Jessen framework to automorphic L-functions. It proves that the distribution of log L(s, sym^k f) on vertical lines converges to a limiting probability measure with a continuous density function, using the Sato-Tate equidistribution and harmonic analysis on tori.

ABSTRACT

We first briefly survey the value-distribution theory of L-functions of the Bohr-Jessen flavor (or the theory of "M-functions"). Limit formulas for the Riemann zeta-function, Dirichlet L-functions, automorphic L-functions etc. are discussed. Then we prove new results on the value-distribution of symmetric power L-functions, which are limit formulas involving associated M-functions.

Motivation & Objective

• To generalize the Bohr-Jessen value-distribution theory to symmetric power L-functions of holomorphic cusp forms.
• To establish limit formulas for the distribution of log L(σ + it, sym^k f) on vertical lines in the critical strip.
• To construct the associated M-function as the weak limit of normalized distribution measures.
• To prove that the limiting density function is continuous and explicitly characterized via harmonic analysis on tori.
• To show that the set of primes for which certain trigonometric conditions hold has positive density, using Sato-Tate equidistribution.

Proposed method

• Adapts the classical Bohr-Jessen method by constructing a finite truncation of the logarithmic L-function via Euler products.
• Defines a multivariate mapping S_N on the N-dimensional torus T^N using the arguments of local factors of the L-function.
• Uses the Sato-Tate equidistribution of Fourier coefficients to control the distribution of angles θ_f(p) modulo 1.
• Applies the Jessen-Wintner inequality and geometric analysis of sums of planar curves to derive convergence of measures.
• Employs the Weyl equidistribution criterion and trigonometric sum estimates to analyze the measure of sets where |sin((γ+1)θ_f(p))| ≥ η.
• Combines the Sato-Tate conjecture (proven by Clozel, Harris, Shepherd-Barron, Taylor) with explicit integral estimates to show positive density of relevant prime sets.

Experimental results

Research questions

• RQ1 Can the Bohr-Jessen limit theorem be extended to symmetric power L-functions of holomorphic cusp forms?
• RQ2 What is the limiting distribution of log L(σ + it, sym^k f) on vertical lines for σ > 1/2?
• RQ3 Does the associated M-function exist as a weak limit of normalized distribution measures?
• RQ4 What is the structure of the limiting density function for symmetric power L-functions?
• RQ5 How does the equidistribution of Sato-Tate angles influence the value-distribution of symmetric power L-functions?

Key findings

• The limit measure W_σ(R; L) exists and equals the integral of a continuous, non-negative density function F_σ(z; L) over a rectangle R.
• The limiting density function F_σ(z; L) is explicitly constructed via harmonic analysis on the torus and is continuous on C.
• The set of primes p for which |sin((γ+1)θ_f(p))| ≥ η has positive density, specifically asymptotic density 1 − 2ξ/π > 0.
• The density of the set {p : θ_f(p) ∈ I_ℓ} is 1 − 2ξ/π + O((log x)^{-1/8 + ε}), confirming positive density for both odd and even γ.
• The convergence of the distribution of log L(σ + it, sym^k f) to the M-function is established via weak convergence of probability measures.
• The proof relies on the Sato-Tate equidistribution of Fourier coefficients and trigonometric sum estimates to control the measure of angular intervals.