# On properties of the Taylor series coefficients of the Riemann xi function at $s=\frac{1}{2}$

## TL;DR

This paper investigates the properties of Taylor series coefficients of the Riemann xi function at the critical point, providing integral formulas, positivity, and monotonicity results for these coefficients.

## Contribution

It introduces new integral formulas involving Gaussian functions for the coefficients and proves their positivity and decreasing nature, advancing understanding of the xi function's local behavior.

## Key findings

- Coefficients $a_k$ are positive.
- Coefficients $a_k$ are decreasing.
- Integral formulas involving Gaussian functions and polynomials.

## Abstract

We prove some properties about the non-zero Taylor series coefficients $a_k$ of the Riemann xi function $\xi(s)$ at $s=\frac{1}{2}$. In particular, we present integral formulas that evaluate $a_k$ whose integrands involve a Gaussian function and a function we call $L(x;k)$. We use these formulas to show that $a_k$ is positive. We also define a sequence of polynomials $p(x;n)$ which arise naturally from the integral formulas and use them to prove that the coefficients $a_k$ are decreasing.