On Pidduck polynomials and zeros of the Riemann zeta function

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.