[Paper Review] Modeling Convolutions of $L$-Functions

Steven J. Miller, Ralph Morrison|arXiv (Cornell University)|Nov 1, 2010

Analytic Number Theory Research48 references3 citations

TL;DR

This paper verifies the Ratios Conjecture's prediction for the one-level density of zeros of the convolution of Ramanujan's tau $L$-function with quadratic Dirichlet $L$-functions, achieving square-root error term accuracy. It demonstrates that the Ratios Conjecture captures arithmetic-dependent lower-order terms—unlike Random Matrix Theory—while showing Kronecker products fail as a random matrix analogue for such convolutions.

ABSTRACT

A number of mathematical methods have been shown to model the zeroes of $L$-functions with remarkable success, including the Ratios Conjecture and Random Matrix Theory. In order to understand the structure of convolutions of families of $L$-functions, we investigate how well these methods model the zeros of such functions. Our primary focus is the convolution of the $L$-function associated to Ramanujan's tau function with the family of quadratic Dirichlet $L$-functions, for which J.B. Conrey and N.C. Snaith computed the Ratios Conjecture's prediction. Our main result is performing the number theory calculations and verifying these predictions for the one-level density for suitably restricted test functions up to square-root error term. Unlike Random Matrix Theory, which only predicts the main term, the Ratios Conjecture detects the arithmetic of the family and makes detailed predictions about their dependence in the lower order terms. Interestingly, while Random Matrix Theory is frequently used to model behavior of L-functions (or at least the main terms), there has been little if any work on the analogue of convolving families of L-functions by convolving random matrix ensembles. We explore one possibility by considering Kronecker products; unfortunately, it appears that this is not the correct random matrix analogue to convolving families.

Motivation & Objective

• To investigate how well the Ratios Conjecture and Random Matrix Theory model the zeros of convolved families of $L$-functions.
• To verify the Ratios Conjecture's prediction for the one-level density of the convolution of Ramanujan's tau $L$-function with quadratic Dirichlet $L$-functions.
• To analyze the arithmetic structure in lower-order terms of the one-level density, contrasting with Random Matrix Theory's focus on main terms.
• To explore whether Kronecker products of random matrix ensembles serve as a valid analogue for convolving $L$-function families.
• To compute and bound error terms in the explicit formula for the one-level density, achieving square-root error term accuracy.

Proposed method

• Uses the explicit formula to relate sums over zeros of $L$-functions to sums over primes and arithmetic functions.
• Applies the Ratios Conjecture's prediction for the one-level density, derived by Conrey and Snaith, to the family of quadratic twists of the tau $L$-function.
• Employs the Fourier transform of test functions and the Mellin transform to analyze the one-level density via contour integration.
• Applies Deligne's bound on Ramanujan's tau function to control error terms in Euler product expansions.
• Uses the $\frac{L'}{L}$-function approach to analyze the logarithmic derivative of $L$-functions at critical points.
• Performs asymptotic analysis of sums over prime powers using bounds on $\tau^*(p^k)$, leveraging Deligne's theorem for error control.

Experimental results

Research questions

• RQ1 To what extent does the Ratios Conjecture accurately predict the one-level density of the convolution of the tau $L$-function with quadratic Dirichlet $L$-functions?
• RQ2 How do lower-order terms in the one-level density reflect arithmetic invariants of the family, and can the Ratios Conjecture capture this dependence?
• RQ3 Can Kronecker products of random matrix ensembles model the convolution of $L$-function families, as Random Matrix Theory models individual $L$-functions?
• RQ4 What is the size and nature of the error term in the one-level density computation, and can it be bounded to square-root accuracy?
• RQ5 How do the analytic properties of the $\frac{L'}{L}$-function for the symmetric square $L$-function relate to the one-level density?

Key findings

• The Ratios Conjecture's prediction for the one-level density matches the number-theoretic computation up to a square-root error term.
• The Ratios Conjecture successfully captures arithmetic dependence in lower-order terms, such as those involving the symmetric square $L$-function and the Riemann zeta function.
• The error term in the one-level density computation is bounded by $O(Q^{-1/2 + \epsilon})$, confirming the conjecture's accuracy at the square-root level.
• Random Matrix Theory fails to model the lower-order arithmetic terms present in the one-level density, as it only predicts the main term.
• Kronecker products of orthogonal and symplectic matrices do not yield a valid random matrix analogue for convolving $L$-function families.
• The analysis confirms that the $\frac{L'}{L}$-function for the symmetric square $L$-function contributes a non-trivial term to the one-level density, detectable via the Ratios Conjecture but not via Random Matrix Theory.