On Pidduck polynomials and zeros of the Riemann zeta function

TL;DR

This paper links zeros of the Riemann zeta function to solutions of a matrix equation involving Pidduck polynomials, providing a new perspective and explicit formulas, with implications for the zeros' simplicity and the Hilbert-Pólya conjecture.

Contribution

It introduces an explicit formula for solutions using Pidduck polynomials and offers an independent matrix characterization of zeta zeros, connecting to orthogonal polynomial bases.

Findings

Derived an explicit formula for solution coefficients in terms of Pidduck polynomials.

Established Pidduck polynomials as an orthogonal basis with a regularized inner product.

Discussed implications for the simplicity of zeta zeros and the Hilbert-Pólya program.

Abstract

For $1<p<\infty$, we prove that a necessary and sufficient condition for $s$ to be a zero of the Riemann zeta function in the strip $0<\Re s<1$ is that $$\left(\begin{array}{cccccc} 1 & \frac{1}{3} & \frac{1}{5} & \frac{1}{7} & \frac{1}{9} & \cdots \\ -\frac{s}{3} & 1 & \frac{1}{3} & \frac{1}{5} & \frac{1}{7} & \cdots \\ -\frac{s}{5} & -\frac{s}{5} & 1 & \frac{1}{3} & \frac{1}{5} & \cdots \\ -\frac{s}{7} &-\frac{s}{7} & -\frac{s}{7} & 1 & \frac{1}{3} & \cdots \\ -\frac{s}{9} & -\frac{s}{9} & -\frac{s}{9} & -\frac{s}{9} & 1 & \cdots \\ \vdots &\vdots & \vdots &\vdots & \vdots & \ddots\\ \end{array}\right)\left(\begin{array}{c} v_0 \\ v_1 \\ v_2 \\ v_3 \\ v_4 \\ \vdots \\ \vdots \\ \end{array}\right) =0 $$ has a nontrivial solution $\left(v_{k}\right)_{k=0}^\infty$ in $\ell^p$. A similar matrix equation was discovered by K. M. Ball in 2017, but the current paper offers a different (and independent) perspective. In this paper an explicit formula for $v_{k}$ is constructed in terms of Pidduck polynomials. In the process, it is also shown that Pidduck polynomials form an orthogonal basis with respect to an inner product of polynomials $f,g$ whereby we replace in a formal expression "$\sum_{n=1}^\infty (-1)^{n+1}n \overline{f(n^2)} g(n^2)$" the divergent sums "$\sum_{n=1}^\infty (-1)^{n+1}n^{1+2k}$" with their zeta-function regularized values. We also discuss the modification for possible non-simple zeros and conclude with applications to the question of the simplicity of the zeros and a relation to the Hilbert-P\'olya program.