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 $$\left(\begin{array}{cccccc}1& \frac{1}{3}& \frac{1}{5}& \frac{1}{7}& \frac{1}{9}& \cdots \\ -\frac{s}{3}& 1& \frac{1}{3}& \frac{1}{5}& \frac{1}{7}& \cdots \\ -\frac{s}{5}& -\frac{s}{5}& 1& \frac{1}{3}& \frac{1}{5}& \cdots \\ -\frac{s}{7}& -\frac{s}{7}& -\frac{s}{7}& 1& \frac{1}{3}& \cdots \\ -\frac{s}{9}& -\frac{s}{9}& -\frac{s}{9}& -\frac{s}{9}& 1& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)\left(\begin{array}{c}{v}_0\\ v_1\\ v_2\\ v_3\\ v_4\\ ⋮\\ ⋮\end{array}\right)=0$$ 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-Pólya program.

57 pages, 3 figures, references added and typos corrected