[Paper Review] Expansion of Dirichlet L-function on the critical line in Meixner-Pollaczek polynomials

Hiroto Inoue|arXiv (Cornell University)|Dec 3, 2014

Advanced Mathematical Identities8 references3 citations

TL;DR

This paper establishes the uniform convergence of the Meixner-Pollaczek polynomial expansion for the completed Riemann zeta function and Dirichlet L-functions on the critical line, proving that partial sums approximate the function uniformly on compact sets and that zeros of the approximating polynomials converge to the non-trivial zeros of the zeta function. The key contribution is a rigorous spectral bound showing the largest zero of the approximating polynomial grows linearly with degree, supporting a novel approach to the Riemann Hypothesis via orthogonal polynomial expansions.

ABSTRACT

We study the expansions of the completed Riemann zeta function and completed Dirichlet L-functions in Meixner-Pollaczek polynomials. We give the proof of the uniform convergence, the multiplicative structure for the coefficients of these expansions, and a calculation for the coefficients of a completed Dirichlet L-function $\hat{L}(s, χ_{-1})$. Furthermore, we give a boundary for the zeros of the approximating polynomial.

Motivation & Objective

• To establish a new orthogonal expansion framework for the completed Riemann zeta function and Dirichlet L-functions using Meixner-Pollaczek polynomials.
• To prove uniform convergence of the MP-expansion in the critical strip, preserving the functional equation's symmetry.
• To derive explicit bounds on the location of zeros of the approximating polynomials, particularly the largest zero.
• To investigate the distribution of complex zeros in the approximating polynomials as a potential indicator for the Riemann Hypothesis.
• To provide a numerical and analytical foundation for using polynomial approximations to study the zero distribution of L-functions.

Proposed method

• Utilizes the Meixner-Pollaczek polynomials $ q_n^{( u,0)}(it) $ as an orthogonal basis in the weighted $ L^2 $-space with measure $ |̳(it + ν/2)|^2 dt $.
• Applies the Darboux method to derive asymptotic expansions for the polynomials, enabling estimation of coefficients and growth behavior.
• Employs the Gerschgorin disk theorem on a tridiagonal-plus-bottom-row matrix representation of the expansion coefficients to bound the eigenvalues (i.e., zeros of the approximating polynomial).
• Derives recurrence relations and trace identities for the matrix representation of the expansion to compute power sums of zeros and infer growth rates.
• Uses the spectral properties of the matrix $ \mathbf{B}_n $, constructed from MP-coefficients, to analyze the location of the zeros of the partial sums $ S_n(t) $.
• Applies the pseudo-spectral method to estimate the possible location of complex zeros within numerical error bounds, as validated by numerical computation for $ n = 260 $.

Experimental results

Research questions

• RQ1 Does the Meixner-Pollaczek polynomial expansion of the completed Riemann zeta function converge uniformly on compact subsets of the critical strip?
• RQ2 Can the zeros of the approximating polynomials $ S_n(t) $ converge to the non-trivial zeros of $ \Xi(t) $, and if so, how fast?
• RQ3 What is the asymptotic growth rate of the largest zero $ \rho_n^{\max} $ of the approximating polynomial $ S_n(t) $ as $ n \to \infty $?
• RQ4 Do complex zeros emerge in the approximating polynomials, and can their location be reliably estimated despite numerical uncertainty?
• RQ5 Can the spectral structure of the coefficient matrix $ \mathbf{B}_n $ provide a bound on the location of the zeros of $ S_n(t) $?

Key findings

• The MP-expansion of the completed Riemann zeta function $ \Xi(t) $ converges uniformly to $ \Xi(t) $ on every compact subset of the critical strip.
• The partial sums $ S_n(t) $ preserve the functional equation symmetry, satisfying $ S_n(-t) = S_n(t) $, due to the parity of the Meixner-Pollaczek polynomials.
• For the completed Dirichlet L-function $ \widehat{L}(s,\chi_{-1}) $, the MP-coefficients are computed explicitly using results from [A] and [K1], enabling numerical analysis.
• The largest zero $ \rho_n^{\max} $ of the approximating polynomial $ S_n(t) $ satisfies $ C' n \leq |\rho_n^{\max}| \leq C n $ for some positive constants $ C, C' $, indicating linear growth with $ n $.
• Numerical results for $ n = 260 $ show that almost all zeros lie on the real axis, but a set of four complex zeros emerges near the origin, consistent with estimation error bounds from the pseudo-spectral method.
• The square sum of the zeros $ \sum_{k=1}^n \rho_{n,k}^2 $ grows asymptotically as $ \sim \frac{2}{3} n^3 $, confirming that not all zeros lie near the imaginary axis.