Uniform Beta Smoothing ofWidder Sums for the Riemann ξ-Function

Source abstract

We study normalized integrals of the Widder expressions associated with $F_\xi = 2G'/G$, where $G(x) = \xi(1/2 + \sqrt{x})$. Their zero weights are regularized incomplete beta kernels. We prove a two-term zero-count formula with error $O_L(\log x)$, uniformly for integers $1 \le m \le Lx$ and each fixed $L > 0$. The proof controls the effect of replacing complex reflected zeros by their ordinates at the same uniform scale. As a consequence, these integrated expressions recover $N(\sqrt{x})$ to error $O_{a,L}(\log x)$ when $a\sqrt{x} \le m \le Lx$. Every cumulative lower density of simple critical-line zeros transfers to the corresponding smoothed sums; Lamzouri's stated proportion theorem supplies the value $0.6725007\ldots$. Separately, we apply the finite Stieltjes framework of Bondesson--Simon to verified zero information, obtaining an explicit range of global Widder positivity. This application neither improves the verified height nor implies the Riemann hypothesis.